Electromagnetic Field Matter Interactions in Thermoelasic Solids and Viscous Fluids (Lecture Notes in Physics)

Lecture Notes in Physics Editorial Board R. Beig, Wien, Austria W. Beiglböck, Heidelberg, Germany W. Domcke, Garching, ...

Author: Kolumban Hutter | Alfons A.F. van de Ven | Ana Ursescu

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Lecture Notes in Physics Editorial Board R. Beig, Wien, Austria W. Beiglböck, Heidelberg, Germany W. Domcke, Garching, Germany B.-G. Englert, Singapore U. Frisch, Nice, France P. Hänggi, Augsburg, Germany G. Hasinger, Garching, Germany K. Hepp, Zürich, Switzerland W. Hillebrandt, Garching, Germany D. Imboden, Zürich, Switzerland R. L. Jaffe, Cambridge, MA, USA R. Lipowsky, Golm, Germany H. v. Löhneysen, Karlsruhe, Germany I. Ojima, Kyoto, Japan D. Sornette, Nice, France, and Zürich, Switzerland S. Theisen, Golm, Germany W. Weise, Garching, Germany J. Wess, München, Germany J. Zittartz, Köln, Germany

The Lecture Notes in Physics The series Lecture Notes in Physics (LNP), founded in 1969, reports new developments in physics research and teaching – quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge in an accessible way. Books published in this series are conceived as bridging material between advanced graduate textbooks and the forefront of research to serve the following purposes: • to be a compact and modern up-to-date source of reference on a well-deﬁned topic; • to serve as an accessible introduction to the ﬁeld to postgraduate students and nonspecialist researchers from related areas; • to be a source of advanced teaching material for specialized seminars, courses and schools. Both monographs and multi-author volumes will be considered for publication. Edited volumes should, however, consist of a very limited number of contributions only. Proceedings will not be considered for LNP. Volumes published in LNP are disseminated both in print and in electronic formats, the electronic archive is available at springerlink.com. The series content is indexed, abstracted and referenced by many abstracting and information services, bibliographic networks, subscription agencies, library networks, and consortia. Proposals should be sent to a member of the Editorial Board, or directly to the managing editor at Springer: Dr. Christian Caron Springer Heidelberg Physics Editorial Department I Tiergartenstrasse 17 69121 Heidelberg/Germany [email protected]

Kolumban Hutter Alfons A. F. van de Ven Ana Ursescu

Electromagnetic Field Matter Interactions in Thermoelastic Solids and Viscous Fluids

ABC

Authors Kolumban Hutter Bergstrasse 5 8044 Zürich Switzerland E-mail: [email protected] Dr. Ana Ursescu Hügelstrasse 21 64283 Darmstadt Germany E-mail: [email protected]

Dr. Alfons A. F. van de Ven Department of Mathematics and Computer Sciences Technical University Eindhoven P.O. Box 513 5600 MB Eindhoven The Netherlands E-mail: [email protected]

K. Hutter et al. Electromagnetic Field Matter Interactions in Thermoelastic Solids and Viscous Fluids, Lect. Notes Phys. 710 (Springer, Berlin Heidelberg 2006), DOI 10.1007/b 11803263

Library of Congress Control Number: 2006930673 ISSN 0075-8450 ISBN-10 3-540-37239-3 Springer Berlin Heidelberg New York ISBN-13 978-3-540-37239-4 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the authors and techbooks using a Springer LATEX macro package Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper

SPIN: 11803263

54/techbooks

543210

Preface

This book is the second and substantially enlarged edition of the Springer Lecture Notes No. 88: Field Matter Interactions in Thermoelastic Solids, which appeared in 1978 and is out of print for about two decades. Since then, the basic issues addressed by the authors in that book have little changed: Because of the nonunique deﬁnition of the electromagnetic ﬁeld quantities in ponderable bodies, constitutive postulates, e.g., for the stress tensor and other ﬁeld quantities, must adequately be postulated, if two theories aiming to describe the same physical phenomenon yield for the same physical problem the same values of the observables. In the Lecture Notes, equivalence relations were established, which guarantee such equivalences for thermoelastic bodies, but no applications of the complex theory were given. Nevertheless, there was a continuous demand for the book, which was fulﬁlled by producing photocopies. In the meantime, however, the authors continued to work on continuum problems of electromagneto-mechanical interactions, in which the theoretical models or simpliﬁcations were applied to practical problems. A.A.F. Van de Ven (AV) worked with students and postdoctoral fellows for more than two decades on problems of magnetoelastic instabilities, i.e., buckling of ferromagnetic and superconducting beams, plates and more complex structures, and on magnetoelastic vibrations of the same type of structures. In the latter problems, it is speciﬁcally the eigenfrequencies that need to be determined, inclusive of their dependence on the applied magnetic ﬁeld or electric current. K. Hutter (KH) who was not involved with electrodynamics for 10 years, took up electromagnetic continua again about 15 years ago and concentrated on applications in ﬂuids, as well as electrorheology. In this ﬁeld, equivalence of formulations is equally a question of electromagneto-mechanical interactions. Here, the central theme is the postulation of adequate constitutive relations, which achieve the electrorheological eﬀect, namely the transition from low viscous ﬂuid behaviour to very high viscous response when the electric ﬁeld is switched on. The application to plane Poiseuille ﬂow of a theory was developed in a Ph.D. dissertation by W. Eckart, to 2D pipe ﬂow with various arrangements of electrodes along the walls of the bottom and the lid of the 2D channel, which was made by Ana Ursescu (AU). Ursescu is joining us as the third author of this book.

VI

Preface

This then outlines the content of the present book. In its ﬁrst part it contains the material of the ﬁrst edition and is due to AV and KH. Only small changes have been made to this text. Few adjustments were necessary because of the additions that were made. A few supplementing references are given to account for the recent literature. In the applications, the new chapter on Magnetoelastic (In)stability and Vibrations is due to AV and the chapter on Electro-rheological Fluids has been drafted by AU along with KH. The entire text has been screened for consistency and homogenization by AV and KH. The increasing interest in electromagnetic problems in the last decade of the last century manifested itself in the appearance of a new journal in 1990, International Journal of Applied Electromagnetics in Materials (IJAEM), founded by K. Miya from Tokyo, and Richard K.T. Hsieh from Stockholm. The ﬁrst plans for this new journal were made in 1986, during a IUTAMSymposium in Tokyo alongwith Miya, Hsieh, Gerard Maugin, Francis Moon, Junji Tani, and one of the authors, AV. The birth of the new journal was accompanied by a series of ISEM symposia (International Symposia on Applied Electromagnetics in Materials, under the chairmanship of Miya) from 1988 until the present time. The 12th ISEM took place in 2005 in Salzburg, Austria. One proviso to the style of the book should be mentioned. Reading the text is not easy. When developing results we are brief and we often outline the steps as to how a result is reached but do not present any details. Thus, the reader is expected to perform the in-between steps, or perhaps even consult the literature. Such an approach is almost unavoidable in electromagnetomechanical interactions. The computations that are in principle not diﬃcult, but generally involved and long, cannot be presented in detail as it would make the book twice as voluminous, and most likely rather boring over long stretches. We regard this as an acceptable compromise. We wish to thank our sponsors and many of our friends in this ﬁeld who have directly or indirectly contributed to this project. AV thanks the Technische Universiteit Eindhoven for their general support to the research on electromagnetoelastic interactions over many years (from 1975 onwards) and he especially thanks, many students for their contributions in particular M. Couwenberg, P. Rongen, P. Smits, and P. van Lieshout, who essentially supported the research on magnetoelastic instabilities. Moreover, he highly appreciates the cooperation and the many fruitful discussions in this area with A.O. J. Tani, Y. Shindo, K. Miya, Shu-Ang Zhou, B. Marusewski, and J.P. Nowacki. KH thanks the Darmstadt University of Technology and the Deutsche Forschungsgemeinschaft (German Research Foundation) for nearly 20 years of support. He also acknowledges the support of W. Eckart and AU in particular for their Ph.D. and postdoctoral fellowships in which the Electrorheological work presented in this book was created. He also thanks Professor K.R. Rajagopal and M. R˚ uˇziˇcka for their interest in our work on electrorheology.

Preface

VII

We also thank E. Vasilieva and A. Maurer for compiling the text of the ﬁrst edition and helping with the editing of the entire manuscript. Finally our thanks go to Dr. Ch. Caron from the Springer Verlag for his interest in this book and his willingness to publish it. Darmstadt, Eindhoven Spring 2006

K. Hutter A. Ursescu A.A.F. Van de Ven

Preface to the First Edition

The last two decades have witnessed a giant impetus in the formulation of electrodynamics of moving media, commencing with the development of the most simple static theory of dielectrics at large elastic deformations, proceeding further to more and more complex interaction models of polarizable and magnetizable bodies of such complexity as to include magnetic dissipation, spin-spin interaction and so on and, ﬁnally, reaching such magistral synthesis as to embrace a great variety of physical eﬀects in a relativistically correct formulation. Unfortunately, the literature being so immense and the methods of approach being so diverse, the newcomer to the subject, who may initially be fascinated by the beauty, breadth and elegance of the formulation may soon be discouraged by his inability to identify two theories as the same, because they look entirely diﬀerent in their formulation, but are suggested to be the same through the description of the physical situations they apply to. With this tractate we aim to provide the reader with the basic concepts of such a comparison. Our intention is a limited one, as we do not treat the most general theory possible, but restrict ourselves to non-relativistic formulations and to theories, which may be termed deformable, polarizable and magnetizable thermoelastic solids. Our question throughout this monograph is basically; what are the existing theories of ﬁeld-matter interactions; are these theories equivalent, and if so; what are the conditions for this equivalence? We are not the ﬁrst ones to be concerned with such fundamental ideas. Indeed, it was W.F. Brown, who raised the question of non-uniqueness of the formulation of quasistatic theories of magnetoelastic interactions, and within the complexity of his theory, he could also resolve it. Penfield and Haus, on the other hand, were fundamentally concerned with the question how electromagnetic body force had to be properly selected. This led them to collect their ﬁndings and to compare the various theories in an excellent monograph, in which they rightly state that equivalence of diﬀerent formulations of electrodynamics of deformable continua cannot be established without resort to the constitutive theory, but at last, they dismissed the proper answer, as their treatment is incomplete in this regard. For this reason the entire matter was re-investigated in the doctoral dissertation of one of us (K. Hutter), but this work was soon found unsatisfactory and incomplete in certain points, although the basic structure of the equivalence proof as given in Chap. 3 of

X

Preface to the First Edition

this tractate, was essentially already outlined there. Moreover, Hutter was still not able to compare certain magnetoelastic interaction theories so that what he attempted remained a torso anyhow. The diﬃculties were overcome by Van de Ven in a series of letters, commencing in fall of 1975, in which we discussed various subtleties of magnetoelastic interactions that had evolved from each of our own work. The correspondence was so fruitful that we soon decided to summarize our eﬀorts in a joint publication. It became this monograph, although this was not our initial intention. Yet, after we realized that a proper treatment required a presentation at considerable length, we decided to be a little broader than is possible in a research report and to write a monograph, which would be suitable at least as a basis for an advanced course in continuum mechanics and electrodynamics (graduate level in the US). We believe that with this text this goal has been achieved. We must at the same time, however, warn the reader not to take this tractate as a basis to learn continuum mechanics and/or electrodynamics from the start. The fundamentals of these subjects are assumed to be known. Our acknowledgements must start with mentioning Profs. J.B. Alblas (Technological University Eindhoven) and Y.H. Pao (Cornell University). They were the ones who initiated our interest in the subject of magnetoelastic interactions. While performing the research for this booklet and during our preparation of the various draughts we were supported by our institutions, the Federal Institute of Technology, Z¨ urich and the Technological University, Eindhoven, and were, furthermore, encouraged by Prof. J.B. Alblas, ¨ller, Paderborn, Prof. Eindhoven, Prof. D. Vischer, Z¨ urich, Prof. I. Mu H. Parkus, Vienna, Dr. Ph. Boulanger, Brussels and Dr. A. Prechtl, Vienna. The support and criticism provided by them, directly or indirectly, were extremely helpful. We are grateful to these people not only for their keen insight and willingness to discuss the issues with us, but also for their encouragement in general. During the initial stage and again towards the end of the write-up of the ﬁnal draught of this monograph K. Hutter was ﬁnancially supported in parts by the Technological University, Eindhoven, to spend a total of a two months period (September 1976 and April 1978) at its Mathematics Department. Without the hospitality and the keen friendship of the faculty and staﬀ members of this department and especially of Prof. J.B. Alblas and his group, the work compiled in these notes would barely have been ﬁnished so timely. The burden of typing the manuscript was taken by Mrs. WolfsVan den Hurk. It was her duty to transform our hand-written draughts into miraculously looking typed sheets of over 200 pages. Her eﬀort, of course, is gratefully acknowledged. Eindhoven and Z¨ urich in the summer of 1978

K. Hutter A.A.F. Van de Ven

Contents

1

General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Equations of Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The Balance Laws of Mechanics . . . . . . . . . . . . . . . . . . . . 2.2.2 The Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Material Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Entropy Production Inequality . . . . . . . . . . . . . . . . . . . . . . . 2.4 Jump Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Material Objectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 9 9 11 14 16 18 21 26

Part I Equivalence of Diﬀerent Electromagnetic Formulations in Thermoelastic Solids 3

4

A Survey of Electromagneto-Mechanical Interaction Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Scope of the Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Two–Dipole Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 The Two–Dipole Model with a Nonsymmetric Stress Tensor (Model I) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 The Two–Dipole Model with a Symmetric Stress Tensor (Model II) . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The Maxwell–Minkowski Formulation (Model III) . . . . . . . . . . 3.5 The Statistical Formulation (Model IV) . . . . . . . . . . . . . . . . . . . 3.6 The Lorentz Formulation (Model V) . . . . . . . . . . . . . . . . . . . . . . 3.7 Thermostatic Equilibrium – Constitutive Equations . . . . . . . . . 3.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 33 38 39 42 51 55 66 71 77 83

Equivalence of the Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.2 Comparison of the Models I and II . . . . . . . . . . . . . . . . . . . . . . . 90

XII

Contents

4.3 4.4 4.5 4.6 5

6

Comparison of the Models I and III . . . . . . . . . . . . . . . . . . . . . . . Comparison of the Models III and IV . . . . . . . . . . . . . . . . . . . . . Comparison of the Models IV and V . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91 93 96 98

Material Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Material Description of the Two–Dipole Models (Models I and II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Material Description of the Statistical and the Lorentz Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Material Description of the Maxwell–Minkowski Formulation . 5.5 Thermostatic Equilibrium – Constitutive Relations for Energy Flux and Electric Current . . . . . . . . . . . . . . . . . . . . . 5.6 Recapitulation and Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Approach to a Uniﬁed Constitutive Theory . . . . . . . . . . . . . . . .

103 103

Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Linearization of the Lorentz Model . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Motivation for this Choice – Governing Equations . . . . 6.2.2 Decomposition of the Balance Laws . . . . . . . . . . . . . . . . . 6.2.3 Decomposition of the Constitutive Equations . . . . . . . . 6.2.4 Decomposition of the Jump and Boundary Conditions 6.3 Linearisation of the Other Models and Comparison . . . . . . . . . 6.4 The Meaning of Interchanging Dependent and Independent Constitutive Variables in one Formulation . . . . . . . . . . . . . . . . . 6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

147 147 152 152 156 163 169 170

104 117 125 130 132 138

181 193

Part II Applications Magnetoelastic (In)stability and Vibrations Electrorheological Fluids 7

Magnetoelastic (In)stability and Vibrations . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Historical Review of Magnetoelastic Buckling Problems . . . . . 7.3 Ferromagnetic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Classical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Variational Method for Ferromagnetic Systems . . . . . . . 7.3.3 Magnetoelastic Buckling of a Set of Two Soft Ferromagnetic Parallel Rods . . . . . . . . . . . . . . . . . . . . . . . 7.4 Superconducting Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Formulation of Variational Principle for Superconducting Structures . . . . . . . . . . . . . . . . . . . . . 7.4.2 A Set of Two Concentric Superconducting Rings . . . . .

201 201 202 205 205 210 217 223 224 231

Contents

7.4.3 How to Use the Law of Biot and Savard in the Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Some Results for Superconducting Structures . . . . . . . . . . . . . . 7.5.1 Review of Speciﬁc Structures and Some Results . . . . . . 7.5.2 The Combined (Variational Biot-Savard) Method . . . . . 7.5.3 Helical or Spiral Superconductors . . . . . . . . . . . . . . . . . . . 7.5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Magnetoelastic Vibrations of Superconducting Structures . . . . 7.6.1 Scope of this Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 Magnetoelastic Vibrations of a Thin Soft Ferromagnetic Circular Plate in a Uniform Transverse Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.3 Magnetoelastic Vibrations of a Superconducting Ring in its Own Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.4 Variational Principle for Magnetoelastic Vibrations of Superconducting Structures . . . . . . . . . . . . . . . . . . . . . 8

9

Electrorheological Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Governing Equations and Constitutive Framework in Electrorheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 The Electromagnetic Momentum Balance . . . . . . . . . . . 8.2.2 The Electromagnetic Energy Balance . . . . . . . . . . . . . . . 8.2.3 Non-relativistic Approximation . . . . . . . . . . . . . . . . . . . . . 8.2.4 The Total Balance Laws of Electrorheology . . . . . . . . . . 8.2.5 Jump Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.7 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Constitutive Laws for the Cauchy Stress Tensor . . . . . . . . . . . . 8.3.1 Models Proposed in the Literature . . . . . . . . . . . . . . . . . . 8.3.2 Constitutive Laws Used in Our (Numerical) Approach 8.4 Applications: Channel Flow of ERFs . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Formulation of the Problem – Electrodes Flush with the Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Particular Case – Inﬁnitely Long Electrodes . . . . . . . . . 8.4.3 Electrodes of Finite Length . . . . . . . . . . . . . . . . . . . . . . . . 8.4.4 Electrodes with Modiﬁed Shape and Position Relative to the Flow – Experimental Results and Discussion . . .

XIII

235 236 236 238 245 252 256 256

258 271 276 279 279 282 283 285 287 287 293 296 297 298 303 303 311 317 318 322 328 361

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 9.1 Appendix A: On Objectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 9.2 Appendix B: Some Detailed Calculations of the Maxwell–Minkowski Model . . . . . . . . . . . . . . . . . . . . . . . . . 373

XIV

Contents

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 Name Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

List of Symbols

Aij A A, Aα a B, Bi  B M , BiM B L , BiL  B S , BiS a B , Bia ◦

◦

B, B i ◦

a

◦ , Bia

Matrix of (anisotropic) heat conduction Matrix of thermal and electrical conduction Vectorial electromagnetic potential Large semi-axis of elliptical cross-section Magnetic induction ﬁeld, magnetic ﬂux density Magnetic induction in the Maxwell-, Lorentzand statistical formulations Auxiliary magnetic induction ﬁeld Magnetic induction in the rest frame

B B, Bα B, Bα

Auxiliary magnetic induction in the rest frame Lagrangean magnetic induction Lagrangean magnetic induction, used in constitutive models ◦ ◦ 2 2 B =B +O(x˙ /c ), Bi =B i +O(x˙ 2 /c2 ) (e) Electrostrictive constants bαβγδ (m)

bαβγδ b, bα B, Bij BS B0 b C Cw C, Cαβ −1 C −1 , Cαβ cαβγδ , cijkl CM c C dA, dAα

Magnetostrictive constants Perturbed Lagrangean magnetic induction Left Cauchy–Green deformation tensor Biot–Savard method Magnetic induction in the rigid-body state Small semi-axis of elliptical cross-section, mean diameter of two concentric tori Speed of light, thermal constant Speciﬁc heat Right Cauchy–Green deformation tensor Inverse of right Cauchy–Green deformation tensor Elastic constants, elasticities Combined method Speed of light in vacuo Symbol for independent constitutive quantity Lagrangean vectorial surface element

XVI

List of Symbols

da, dai dl, dli dv, dV D, Di DM , DM i D a , Dia

Eulerian vectorial surface element Eulerian vectorial length element Eulerian (Lagrangean) volume element Dielectric displacement Maxwellian dielectric displacement Auxiliary dielectric displacement

D, Di

Dielectric displacement in the rest frame

◦

◦

◦

◦ D a , Dia ◦ Di =Di

Auxiliary dielectric displacement in the rest frame

+O(x˙ /c2 ) D, Dα Lagrangean dielectric displacement Da , Dαa Auxiliary Lagrangean dielectric displacement d, dα Perturbed Lagrangean dielectric displacement da , daα Perturbed Lagrangean auxiliary dielectric displacement D Plate rigidity d Distance of two concentric tori dij Stretching, rate of strain tensor D0 Regularization parameter for the constitutive relation of the extra stress tensor E, Eαβ Lagrangean strain tensor, Green strain tensor e, eij Eulerian strain tensor, Finger strain tensor e, eαβ Perturbed Green strain tensor E, Ei  Electric ﬁeld strength E C , EiC    Electric ﬁeld strength in the Chu-, Maxwell-, E M , EiM L L  Lorentz- and statistical formulations E , Ei   E S , EiS E, Ei Electromotive intensity ◦

E E, Eα eijk eABCD e, eα E ei (i = 1, 2, 3) eij E0 F , Fi F0 f F e , Fie F ext , Fiext

2

Electric ﬁeld strength in the rest frame Lagrangean electric ﬁeld strength,- electromotive intensity Three-dimensional permutation tensor Four-dimensional perturbation tensor Perturbed Lagrangean electromotive intensity Young’s modulus Unit vectors Inﬁnitesimal strain tensor Scale for electric ﬁeld Speciﬁc body force, Biot–Savard force Biot–Savard force in the rigid-body state Biot–Savard force in the perturbed state Electromagnetic body force External body force

List of Symbols

F Lorentz F e , Fαe F ext , Fαext F  F∆p (Q) F  f1 , f2 , f3 G, Gα g, gi G G, Gαβ G GI GY H, Hi H C , HiC H M , HiM H a , Hia Ha , Hai ◦

◦

H, Hi H, Hα Ha , Hαa H, Hα H AB h, hα ha , haα h I IAB Iy J J j, jα

 j1αβγ  j2αβ , j3αβ , j5αβ  j4α

XVII

Lorentz force Total electric force due to surface tractions of the Maxwell stress tensor just outside the body Total force on a body due to external body force and surface traction distributions Deformation gradient Quantities measuring the electrorheological eﬀect Lagrangean electromagnetic momentum density Eulerian electromagnetic momentum density Elastic shear modulus Lagrangean strain tensor (Green strain tensor) Shear modulus Intermediate state Bending moment in a ring Magnetic ﬁeld strength Magnetic ﬁeld strength in the Chu- and Maxwell formulations Auxiliary magnetic ﬁeld strength Auxiliary eﬀective magnetic ﬁeld strength Magnetic ﬁeld strength in the rest frame Lagrangean eﬀective magnetic ﬁeld Auxiliary Lagrangean eﬀective magnetic ﬁeld strength Lagrangean magnetic ﬁeld-strength, used in the constitutive relations Magnetic ﬁeld-electric displacement four-tensor Perturbed Lagrangean eﬀective magnetic ﬁeld Perturbed Lagrangean auxiliary eﬀective magnetic ﬁeld Gap width Thermodynamic potential in determining the internal energy from electromagnetic integrability conditions Moment of inertia of the body relative to its center of mass in the rigid-body state Moment of inertia of an area Jacobian determinant, determinant of the deformation gradient Thermodynamic potential, determining the contribution of the deformation to the internal energy Perturbed Lagrangean electric current ﬁeld Coeﬃcients arising in the constitutive relation for the Lagrangean conductive electric current

XVIII List of Symbols

J

Homogeneous quadratic functional of the perturbation ﬁelds of which δJ is the second variation of the action functional L Bessel functions of zeroth and ﬁrst order

J0 , J1 J , J e, J m Ji , Jie , Jim J , Ji J e , Jie , J m, J m i , ◦

◦

J,Ji

Non-conductive current density,-electric, -magnetic current density Conductive current density Electric, magnetic current densities Conductive current in the rest frame

 

J, Jα Je , Jeα  Jm , Jm α

Lagrangean conductive current density Lagrangean electric (magnetic) conductive current density

J K k L L(e) L(m) L L, Li L, Li L, Lij Lext , Lext i Le , Leα

Common factor of J Magnetic part of J, K = B02 K, K = I02 K Longitudinal pressure gradient Characteristic length Thermoelectric constant Thermomagnetic constant Action integral Lagrange densities Speciﬁc body couple, Dual tensor of the body couple vector Dual tensor of the external body couple vector Total electromagnetic couple relative to the center of gravity in the rigid-body state due to the Maxwell stress tensor just outside the body Total external couple due to body couples, body forces and surface tractions

Lext , Lext α  M C , MiC    M M , MiM M L , MiL    M S , MiS M, Mi MM , MM i M, Mα M, Mα m

Magnetization density in the Chu-, Maxwell- , Lorentz- and statistical formulations

Magnetization density, – in the Maxwell formulation in the rest frame Lagrangean magnetization density Lagrangean magnetization density per unit mass Viscosity scale for the power law constitutive relation for the extra stress tensor

List of Symbols

 

m, mα mC , mC α  mL , mL α

XIX

Perturbed magnetization, – in the Chu- and Lorentz formulations

 m1αβγ  m2αβ , m3αβ , m5αβ Coeﬃcients arising in the constitutive relation for the  perturbed magnetization m4αβ Mrr Ma n, ni N , Nα n, n0 , n∞ N1 , N2 nAB (A, B = 1, 2, 3) O, Oij P , Pi  P C , PiC  P M , PiM  P S , PiS P, Pα P, Pα p, pα  p1αβγ  p2αβ , p3αβ  p4α p q q q, qα q, qi q S , qiS Q, Qα Q Q, Qe , Qm ◦

Q Q, Qe , Qm

 q1αβγ  q2αβ , q3αβ , q5αβ  q4α Q Qj (j = 1, .., n)

Radial bending moment in a ring-like structure Mason number Eulerian unit normal vector Lagrangean unit normal vector Exponents in the power law of the extra stress tensor Normal stress diﬀerences Coeﬃcients in the parameterization of ηgen Orthogonal 3 × 3 matrix Polarization density Polarization density in the Chu-, Maxwell- and statistical formulations Lagrangean polarization density Lagrangean polarization density per unit mass Perturbed Lagrangean polarization density Coeﬃcients arising in the constitutive relation of the perturbed polarization Pressure Transverse load of a beam or plate Perturbed Lagrangean charge density Perturbed Lagrangean energy ﬂux Energy (heat) ﬂux vector Energy (heat) ﬂux vector in the statistical formulation Lagrangean energy (heat) ﬂux vector Electric charge density, surface charge density Charge density, electric-, magneticCharge density in the rest frame Lagrangean charge density, electric-, magnetic-. Coeﬃcients arising in the constitutive equation for the perturbed Lagrangean energy ﬂux Quantity in the evaluation of I0 Contribution to the magnetic transverse load of beams and plates

XX

List of Symbols

Q r re rext Rαβ R r 1 , r2 Re s s  s1αβ  s2α , s3α  s4 S S1 S0 Si Si0 si T ,T 0 T1 Ty T T , Tiα M Tiα p T p , Tαβ ti (i = 1, 2) teij t, tij tM , tM ij (n)

t(n) , tij t, tiα tαβ δiα tαβ  t1αβγδ  t2αβγ , t3αβγ  t4αβ U ¯ U u

Discharge Energy supply density Electromagnetic energy supply density External energy supply density Orthogonal matrix, describing the rotation of a rigid-body Radius of a circular cross-section Position vectors on wires, –1,2 Reynolds number Entropy supply density Perturbation entropy density Coeﬃcients arising in the constitutive relation for the perturbed entropy density Set of equations characterizing the equilibrium state of a magnetoelastic system subject to an external ﬁeld Analogous set as S describing the perturbed state Analogous set as S describing the rigid-body state Element of S Element of S 0 Element of S 1 Stress tensor, - in rigid-body state Perturbed stress tensor Traction in the y-direction along the boundary of the cross-section Kinetic energy First Piola–Kirchhoff stress tensor First Piola–Kirchhoff–Maxwell stress tensor Second Piola–Kirchhoff stress tensor Unit vector tangential to the i-th inﬁnitely thin wire Extra Cauchy stress tensor Cauchy stress tensor Cauchy–Maxwell stress tensor Stress vector, traction Perturbed ﬁrst Piola–Kirchhoff stress tensor Coeﬃcients arising in the constitutive relation of the perturbed ﬁrst Piola–Kirchhoff stress tensor Internal energy density Particle displacement from the initial to the intermediate conﬁguration Particle displacement from the intermediate to the ﬁnal state

List of Symbols

XXI

U u V , Vi V v VM V0 V v1 v2 WN wn WM W

Helmholtz free energy, internal energy Elliptical coordinate Velocity, arising in the Lorentz transformation Transverse displacement in beam and plate bending Elliptical coordinate Variational method Scale for material velocity Electric potential Longitudinal velocity Transverse velocity Speed of propagation Speed of displacement Maxwell modiﬁed internal energy density Bending energy of beams or plates, part of the elastic energy independent of B w(m) (z) = Complex displacement (m) u(m) (z) + iv (z) x Position vector xi X, Xα Particle position in reference conﬁguration 2 Z = (D02 + γ˙2 ) Regularized second stretching invariant in simple shear ﬂow Greek Symbols α1 , . . . , α6 αAB (A = 1 , . . . , 6; B = 1, 0) ˆ βi (i = 1, 2) β β1 , β2 βαβ βij γ γ˙ γ = 0.5772 δ δJ δL δαβ , δij ε ε = πa/(4l)

Scalar, parameters in the isotropic representation of the extra stress tensor Scalar, parameters in the isotropic representation of the extra stress tensor Parameters in the constitutive equation for the shear stress Thickness to width ratio Parameters arising in the Casson-like constitutive function Coupling matrix of thermal and electrical conduction Entropy production density Shear rate in simple shear ﬂow Euler number Regularization parameter in the constitutive relation for stress Second variation of the action integral L First variation of the action integral Kronecker delta Eﬀective permittivity Slenderness parameter

XXII

List of Symbols

ε ε0 ε˜, ε¯ (e) εαβγ (m)

εαβγ ξ, ξα Ξ, Ξα ξ Ξ Ξ η η AB η0 , η∞ ηgen Θ θ κij , καβ

Regularization parameter in the constitutive relation for stress Dielectric constant in vacuo, electric permittivity of the free space Legendre transformed internal energy Piezoelectric constants Piezomagnetic constants Particle position in intermediate state Center of mass in the rigid-body state Parameter to regularize the power law term of the extra stress tensor at zero electric ﬁeld General physical quantity Displacement from the initial to the intermediate state Entropy density Magnetic ﬁeld strength-electric displacement four-tensor Viscosities in the bi-viscosity models Generalized viscosity Absolute temperature Perturbed temperature Eulerian, Lagrangean matrix of heat conductivity

κ = −γ − ln (1 + β)/2

Λ = 1 + (2βµε2 κ)−1 ΛAB Matrix of extended Lorentz group ´ constant, eigenvalue of a matrix λ Lame (e) λα Pyroelectric constant (m) Pyromagnetic constant λα µ0 Magnetic permeability in free space µ Magnetic permeability µAB Magnetization four-tensor ν Thermoelastic constant ναβ Thermoelastic coupling tensor ν Poisson number πα , πi Electromagnetic energy ﬂux vector π AB (Magnetization)-polarization four-tensor ρ Mass density in present conﬁguration ρ0 Mass density in reference conﬁguration Σ, σ Singular surface σA Electric current, electric charge density four-vector σαβ Matrix of electric conductivity σA (A = 1, 2, 3) Eﬀective electric conductivities σABC Eﬀective electric conductivity constants Φ Physical variable (unspeciﬁed)

List of Symbols XXIII

Φ, Φi Φ, Φα Φ(0) Φj ϕAB ϕ (m) 2χ (e) 2χ (e) 4χ (m) 4χ ψ ψ¯0,1,2

Entropy ﬂux (Eulerian) Lagrangean entropy ﬂux, electromagnetic scalar potential Magnetic potential in the rigid-body state General ﬂux quantity Electric ﬁeld-magnetic induction four-tensor Perturbed magnetic potential Second-order magnetic susceptibility Second-order electric susceptibility Fourth-order electric susceptibility Fourth-order magnetic susceptibility External perturbed magnetic potential Parameters in the constitutive relation for the normal stress components Helmholtz free energy density Legendre transform of ψ Legendre transformed ψ Legendre transformed ψ Covariant four-tensor (e.g. for polarization) Contravariant four-tensor (e.g. magnetization) Shear stress of the Bingham, Casson and Casson-like model Yield stress in the Bingham and classical Casson model Electromagnetic energy density Angular velocity in the rigid-body motion Electromagnetic speciﬁc energy density

ψ ψ˜ ψ¯ ψˇ ψ AB ψAB τ B, τ C, τ U τy Ω, ω Ω, Ωα ω

Other Symbols A O ( )˙

Convective derivative Order symbol Total or material time derivative

d = dt + f = f − f − Jump of f at a singular surface f = 12 (f + + f − ) f(ij) Symmetric part of fij f[ij] Skew-symmetric part of fij ∗

fi fE f¯ F0

f

Corotational derivative of fi f , evaluated in thermostatic equilibrium f¯, evaluated in the intermediate state Norm of f , usually lim sup | f | F , evaluated in the rigid-body state

1 General Introduction

With the boundaries of physical science expanding rapidly, present day engineers must necessarily assimilate information and knowledge of subjects that continue to become more and more complex. In the study of the nature and mechanical behavior of engineering materials, however, their knowledge seldom progresses beyond the level of elementary theory of elasticity and viscous ﬂuids and the basic concepts of the theory of plasticity and viscoelasticity. More esoteric theories of material behavior and the interaction with various ﬁelds are generally left out of consideration or soon abolished as being mathematically intractable or economically unjustiﬁable. This book is an account on one of the above mentioned more esoteric theories. What we have in mind is the response of deformable bodies (solids or ﬂuids) to electromagnetic ﬁelds. Indeed, the interaction of electromagnetism with thermoelastic ﬁelds is not only a challenging scientiﬁc problem, but it is increasingly attracting also engineers in the materials sciences and in nanotechnology from a fundamental and in the nuclear power and electronic industry from a purely applied point of view. The response of deformable bodies to electromagnetic ﬁelds is diﬃcult to describe for several reasons. First, it is fundamentally nontrivial because the formulation of electrodynamics in ponderable bodies is nonunique. In other words, one generally introduces four diﬀerent electromagnetic vector ﬁelds, e.g. the electric ﬁeld strength E and the magnetic induction B plus two other ﬁelds. Some authors work with electric displacement D and magnetic ﬁeld strength H, others use polarization P and magnetization M instead, but there are several variants how to introduce the latter two quantities. This as such would not make yet electrodynamics of deformable media fundamentally diﬃcult, as only unique transformations from the variables (E, D, B, H) to (P , M ) would be needed. The diﬃculty arises with the deﬁnitions of the electromagnetic stress tensor and speciﬁc body force in the momentum equation (and in theories with polar structure corresponding deﬁnitions of the electromagnetic couple stress and speciﬁc body couple). Consequently, for two scientists using diﬀerent electromagnetic variables and seemingly unmatchable expressions for stress and speciﬁc body force, but aiming to describe the same electromagneto-mechanical phenomenon, there arises the question, whether the two formulations can ever yield the same results for the K. Hutter et al.: Electromagnetic Field Matter Interaction in Thermoelastic Solids and Viscous Fluids, Lect. Notes Phys. 710, 1–6 (2006) c Springer-Verlag Berlin Heidelberg 2006 DOI 10.1007/3-540-37240-7 1

2

1 General Introduction

measurable quantities in the mathematical reproduction of the same physical situation. In Part I of this book we shall give an answer to this diﬃcult problem. The form of the answer will demonstrate that thermodynamic arguments involving the Second Law of Thermodynamics are needed to resolve this fundamental question. The analysis presented in Part I will be performed on the level of nonrelativistic approximation. This is a matter of judicious compromise, as our interest is in technical applications for which mechanical equations are classical and Galilean. Alternatively, the Maxwell equations are Lorentz invariant. To merge the two, an approximation will be introduced that is, in electromagnetic variables and combinations thereof, accurate up to terms of order O(v 2 /c2 ), which will be omitted. Here, v denotes the modulus of the material velocity and c is the speed of light. Second, the description of deformable bodies to electromagnetic ﬁelds is also diﬃcult, because the formulation of rather simple initial boundary value problems leads, in general, to rather complex solution procedures, i.e., the equations are generally structurally coupled and complex, i.e. nonlinear. When deformations of solid bodies are concerned, then it is often so that the deformations from the initial to the ﬁnal positions are small. Under such circumstances one may compose the total response to an external magnetic ﬁeld of an intermediate conﬁguration that is known plus a correction from it that is small, so that linearization of the perturbed equations is justiﬁed. In many situations it is justiﬁed to identify the intermediate conﬁguration with the initial conﬁguration, also called rigid-body state. The electromagnetic ﬁelds as a response to the external driving ﬁelds are then in a ﬁrst step calculated for the rigid-body state and then corrected for the deformation. In these perturbed equations the perturbed electromagnetic ﬁelds are coupled with the mechanical equations for the displacement ﬁeld. They are nonlinear but on the prerequisite that the displacements and the corrections of the electromagnetic ﬁelds due to these displacements are equally small, these equations may be linearized. This makes the analysis much simpler, even though for many problems analytical solutions still cannot be found. We shall demonstrate in the last chapter of Part I how this decomposition can systematically be implemented. Part II of this book is devoted to two diﬀerent, but technically important engineering applications. Here, the complexity of the problem formulation may be simply due to the surprising subleties of the electromagnetomechanical interactions that require adjustment of the novel reader’s attitude to the delicacy of the results. We will shortly give an example. We shall treat two diﬀerent examples. The ﬁrst concerns the reaction of beams or plates to an external magnetic ﬁeld. Early experiments, performed in the 60s of the last century and theories describing the buckling of ﬂat and wide cantilever beams did not yield the same functional relation for the buckling value of the external magnetic ﬁeld B0 . The reason was found more than

1 General Introduction

3

20 years later. On the basis that the ratio “beam thickness to beam width”, h/w 1, was small, it was concluded that the ﬁniteness of the width of the beam could be ignored and w → ∞ or h/w = 0 could be assumed. This is too simple, because for the perturbed magnetic potential, satisfying a 2D Helmholtz equation and not a 2D Laplace equation, it is not allowed to approximate the narrow rectangular cross-section by an inﬁnitely wide one (no matter how small the ratio h/w becomes). Once this was recognized and the perturbed magnetic ﬁelds were calculated on the basis of the correct geometry of the beam, experimental and theoretical analytic results could be brought into coincidence. An analytical solution to this problem was found for a Bernoulli beam with elliptical cross-section and arbitrary aspect ratio. Rectangular cross-sections face diﬃculties because of singularities that develop in corners. One procedure in the determination of the magnetoelastic critical magnetic ﬁeld is to start from the homogeneous set of linear equations with homogeneous boundary conditions for the perturbed ﬁelds and to determine the smallest value for the magnetic ﬁeld in the rigid-body state that allows for a nontrivial solution. This is a linear eigenvalue problem and the parameter characterizing this critical magnetic ﬁeld is the eigenvalue of this linear boundary value problem with nontrivial eigenvector. For ferromagnetic materials this linear problem is replaced by a variational formulation with higher-order accuracy. This variational formulation is applied to the magnetoelastic response of two parallel circular rods, to a superconducting torus and sets of two concentric tori and a pair of tori with circular cross sections. The solution of the linear elliptic partial diﬀerential equations requires use of conformal mapping techniques and complex variable theory, which, in these times of immediate application of electronic computation are often considered to be diﬃcult, even though they belong to the classical syllabus of the education in theoretical engineering, physics and applied mathematics. Another approximation that has been popular in determining the critical current in superconducting structures is the application of the law of Biot–Savard. This law provides a formula for the electromagnetic force in an inﬁnitely thin wire due to a second inﬁnitely thin wire, both carrying an electric current. For parallel wires, ﬁnite and inﬁnite helical wires the Law of Biot–Savard allows determination of the bucklung current. Application of the variational method or a combined method using the variational formulation and the law of Biot–Savard shows that the Biot–Savard approach can only yield reliable results, provided the distance of the wires or of consecutive tori is large as compared to the diameter of the wire. Interesting and somewhat counterintuitive is the result that the buckling current of an inﬁnitely long helical wire is never approached by the buckling current of a helical wire with n → ∞ coils. The second example of electro-mechanical interactions is provided in the last chapter where electrorheological ﬂuids are treated. These are actually

4

1 General Introduction

mixtures of an electrically neutral ﬂuid with suspended polarizable particles. Phenomenologically, when the electrical ﬁeld is switched oﬀ the particles, which are of subgrid scale, are randomly distributed and the mixture, treated as a ﬂuid, appears to have small viscosity. Alternatively, when the electric ﬁeld is switched on, the polarizable particles arrange in columns more or less perpendicularly to the ﬂow and so stiﬀen the motion against shearing. This becomes manifest as a substantial increase in viscosity almost to the extent that the mixture now behaves much like a solid. The theoretical description of electrorheological ﬂuids as we employ it is based on a single constituent continuum concept and postulations of the electromagnetic stress tensor, speciﬁc body force, energy supply and energy ﬂux. These are derived from the Maxwell equations in the Maxwell– Minkowski formulation by deriving the electromagnetic balance laws of momentum and energy. What emerges are e.g. two forms of electromagnetic stresses and corresponding speciﬁc electromagnetic forces. By a scaling analysis these are reduced to their nonrelativistic counterparts. The constitutive theory is more complex than the corresponding theory in Part I, because the mechanical stress tensor depends not only on the density, temperature, electric ﬁeld but also on the rate of strain tensors. The diﬃculty in arriving at a theoretical formulation for electrorheological ﬂuids is in a proper three-dimensional formulation of the constitutive relation for the “mechanical” stress tensor and how it matches experiments, mostly in two-dimensional Poiseuille-type ﬂows such that the Second Law of Thermodynamics is satisﬁed. This leads to Bingham-type and power-law type stress-stretching relations and extensions thereof in which the dependence on the electric ﬁeld is particularly subtle. The optimal model is subsequently applied to two-dimensional channel ﬂows of which the walls are alternatively covered by inﬁnitely thin charged electrodes or left uncovered. The electric ﬁeld that would be uniform, if electrodes were inﬁnitely long, will now be inhomogeneous. In particular at electrode ends square root singularities develop, and if electrodes are not long in comparison to the gap width, the local electric ﬁelds will mutually interact. Of interest is, how these inhomogeneities aﬀect the discharge through the two-dimensional channel or how the pressure drop between two positions along the channel axis will depend on the relative electrode lengths and their gap widths. Additional questions are, how ﬁnitely thick electrodes will affect the pressure drop, if they conﬁne or enlarge the gap width or yield a converging or diverging channel stretch. Such questions are also asked when electrodes are corrugated or beamed and thus destroy the nearly viscometric ﬂow of the ﬂuid. This is roughly speaking the content of the book. After the appearance of the ﬁrst edition of this book, the interest in electromagnetomechanical problems increased strongly. This was manifested for instance by the start of a new journal International Journal of Applied

1 General Introduction

5

Electromagnetics in Materials (IJAEM) in 1990, edited by K. Miya and R.K.T. Hsieh. In 1995, the journal changed its name slightly in International Journal of Applied Electromagnetics and Mechanics with K. Miya as editor-in-chief. Parallel with this journal a series of symposia called ISEM (International Symposium on Applied Electromagnetics in Materials) found place every 2 years, the ﬁrst being held in 1988 in Tokyo and the up to now last one in 2005 in Salzburg, Austria. The proceedings of these symposia appeared, besides in [189] and [190], as special volumes of IJAEM. For instance, the proceedings of the 2003-symposium in Versailles, France, can be found as a special issue in [191]. During this period, also, several new books were published, of which we here only mention the works of Eringen and Maugin [70, 71], Maugin et al. [137], Tiersten [238], and Shu-Ang Zhou [215, 216]. Besides the two ﬁelds of applications dealt with in the Chaps. 7 and 8 of this book, a lot of other ﬁelds exists in which electromagneto-mechanical models are applied. Far from aiming to be general, we mention here as examples the following three subjects: Piezoelectric Devices This subject is so diverse and knows so many diﬀerent applications, such as non-destructive testing by means of sonar, ignition lighters, ultrasonic transducers in remote controls, transducers in ink-jet printers and many more, that a complete review is far beyond the scope of this book. Also, the amount of literature appearing yearly on this subject is immense. Therefore, we restrict ourselves here to mentioning only the classical treatments by Tiersten [235], Ikeda [101] and Taylor [232], and two, more recent ones, by Ding and Chen [58] on 3D problems piezoelasticity, and by Qin [193] on fracture in piezoelectric materials. Finally, Kamlah [106] gives an account on ferroelectric and ferroelastic piezoceramics. Cracks in Magnetoelastic or Piezoelectric Plates The investigation of crack propagation and the (non-destructive) detecting of cracks in magnetoelastic or piezoelectric materials is an important subject in fracture mechanics. One of the authors having a great impetus in this respect is certainly Y. Sindo, who published since 1981, [211], up to now on cracks in magnetoelastic or piezoelectric layers and plates; see e.g. [212] and [213] and, of more recent date, [125] and [214]. Some very recent papers are by Hasanyan and Piliposian [87], Y. and I. Podil’chuck [183], Li and Yang [119], and Wang and Jiang [262]. Waves in Electromagneto-Elastic Solids This is a very classical subject that received attention from the ﬁrst beginnings of the research on electromagneto-elastic interactions by the founding

6

1 General Introduction

fathers of this theory such as Pao, Parkus, Eringen, Maugin and many others. We restrict ourselves to giving only a few more recent references, which consider successively surface waves in magnetoelastic conductors, Hhefni et al. [88], waves at piezoelectric interfaces, Romeo, [201, 202] and Jang and Zhou, [103] and ﬂexure waves in electroelastic plates by Lancioni and Tomasetti [117].

2 Basic Concepts

In this chapter we shall lay down the foundations of electrodynamics and continuum mechanics as it will be used in this book. We start with kinematics and then turn to the equations of balance, the Maxwell equations and the balance laws of mass, momenta, energy and entropy. We introduce the concept of material objectivity in the nonrelativistic approximation and brieﬂy outline constitutive equations of thermoelasticity.

2.1 Kinematics As is common in continuum mechanics, we regard a body as a threedimensional manifold embedded in Euclidian 3-space. Its elements are called particles. Let RR be its reference conﬁguration and Rt its conﬁguration at time t. Instead of Rt we shall subsequently write R, and we shall refer to R as the present conﬁguration of the body. Parts of the body will be denoted by VR and V, dependent on whether they are referred to the reference conﬁguration and the present conﬁguration, respectively. The boundary of VR and of V will be denoted by ∂VR and ∂V, respectively. The position of a particle in RR will be designated by X(Xα , α = 1, 2, 3), whereas that in the present conﬁguration R is x (xi , i = 1, 2, 3). A motion of the body is then described by the mapping (i, α = 1, 2, 3) . (2.1.1) xi = χi (Xα , t) , We assume χ to be invertible and this is tantamount to assuming that the functional determinant (2.1.2) J := det (∂χi /∂Xα ) , never vanishes and may without loss of generality be assumed to be positive. In the above and throughout this monograph symbolic and Cartesian tensor notation is used. Greek indices refer to the material coordinates X, and Latin indices to the spatial coordinates x. Summation convention will be used over doubly repeated indices and commas preceding indices indicate diﬀerentiations with respect to space variables. If Aij denote the components of a second rank tensor, then A(ij) = 12 (Aij + Aji ) ,

A[ij] = 12 (Aij − Aji )

(2.1.3)

K. Hutter et al.: Electromagnetic Field Matter Interaction in Thermoelastic Solids and Viscous Fluids, Lect. Notes Phys. 710, 7–29 (2006) c Springer-Verlag Berlin Heidelberg 2006 DOI 10.1007/3-540-37240-7 2

8

2 Basic Concepts

denote its symmetric and skew-symmetric parts such that Aij = A(ij) + A[ij] .

(2.1.4)

Moreover tr A = Aii will denote its trace. The symbol d/dt or the superimposed dot will designate diﬀerentiation with respect to time t, holding the ˆ particle X ﬁxed, i.e. for Φ = Φ(X, t) ˆ ∂ Φ(X, t) dΦ ≡ Φ˙ := . dt ∂t

(2.1.5)

Here, Φ˙ is called the material time derivative. Likewise, ∂/∂t will denote diﬀerentiation with respect to time t, holding the spatial position x ﬁxed. ˜ t), Hence, with Φ = Φ(x, ˜ t) ∂Φ ∂ Φ(x, := , (2.1.6) ∂t ∂t where ∂Φ/∂t is called the local, or partial, time derivative. It is easy to see that (2.1.5) and (2.1.6) may be combined to yield dΦ ∂ Φ˜ ∂ Φ˜ = Φ˙ = + x˙ i . dt ∂t ∂xi

(2.1.7)

The local deformation of a body in the neighborhood of a particle X may be characterized to ﬁrst order by the deformation gradient Fiα , deﬁned by Fiα :=

∂χi (X, t) = xi,α . ∂Xα

(2.1.8)

Its inverse exists and is written as −1 Fαi = Xα,i .

(2.1.9)

Of particular interest are objective combinations of Fiα . The right (left) Cauchy-Green deformation tensors Cαβ , (Bij ) are deﬁned by Cαβ := Fiα Fiβ ,

Bij = Fiα Fjα .

(2.1.10)

In applications it is more convenient to use the Lagrangean (Eulerian) strain tensors, Gαβ , (Eij ) or the deformation tensors, Gαβ := 12 (Cαβ − δαβ ) , instead. Here,

δαβ =

Eij := 12 (δij − Bij )

1α=β, 0 otherwise

(2.1.11)

(2.1.12)

is the Kronecker delta. The velocity of a material particle is the time rate of change of its position. In the Lagrangean formulation it takes the form

2.2 Equations of Balance

x˙ i =

∂χi (Xα , t) = vˆi (Xα , t) . ∂t

9

(2.1.13)

The Euler representation of the velocity reads x˙ i = vˆi (Xα−1 (xn , t), t) = v˜i (xn , t) .

(2.1.14)

The time rate of change of the velocity of a material element is its acceleration. We express this in the Euler representation as x ¨i = v˙ i =

∂˜ vi d˜ vi (xn , t) = + Lij v˜j , dt ∂t

(2.1.15)

in which Lij is the spatial velocity gradient deﬁned as ˙ ij . Lij = x˙ i,j = (grad x)

(2.1.16)

Its symmetric, dij , and skew-symmetric, Wij , parts dij := L(ij) ,

Wij := L[ij] ,

(2.1.17)

are called stretching or rate of strain tensor and vorticity or spin tensor, respectively. Combining (2.1.8) with (2.1.14) we see that F˙iα = (xi,α ) = vˆi,α v˜i,j Fjα = Lij Fjα , −1 or after multiplication with Fαk −1 Lik = F˙ iα Fαk .

(2.1.18)

This formula expresses the spatial velocity gradient in terms of the material deformation gradient.

2.2 Equations of Balance 2.2.1 The Balance Laws of Mechanics It is the ultimate goal of any thermodynamic theory, be it a theory of a simple ﬂuid or a fairly complicated description of electromechanical interaction phenomena, to calculate within a body the independent ﬁelds of this theory as functions of space and time. For the determination of these ﬁelds, we need ﬁeld equations, which are obtained when the balance laws of mechanics and electrodynamics are combined with constitutive equations. Basic ingredients of any theory are thus the balance equations which are discussed below. We start with the balance laws of mechanics. The integral expressions of the laws of conservation of mass, balance of momentum, moment of momentum and energy in the spatial and material description are well-known. For the former they are

10

2 Basic Concepts

d dt d dt d dt and d dt

ρx˙ i dν = V

(2.2.1)

tij daj + ∂V

ρFi dν ,

(2.2.2)

ρ(Lij + x[i Fj] )dν ,

(2.2.3)

V

x[i tj]k dak + V

∂V

( 12 ρx˙ i x˙ i V

ρdν = 0 , V

ρx[i x˙ j] dν = V

(x˙ i tij − qj )daj +

+ ρU )dν =

(ρx˙ i Fi + ρr)dν .

(2.2.4)

V

∂V

Here, ρ is the mass density per unit volume, tij is the Cauchy stress tensor, Fi the total body force due to electromagnetic ﬁelds and external actions, Lij the dual of the body couple Lk , i.e. Lij = 12 eijk Lk ,

(2.2.5)

U is the internal energy per unit mass, qi the energy ﬂux, consisting of heat ﬂux and non-thermal energy ﬂux, and r the total energy supply, due to the electromagnetic ﬁelds and due to heat. Bracketed indices indicate anti(n) symmetric tensors. Note that the stress tensor tij and the stress vector ti on a surface element with exterior unit normal n are related by (n)

ti

= tij nj .

(2.2.6)

Often, in the literature the stress tensor is deﬁned through (n)

ti

= t¯ji nj .

(2.2.7)

This deﬁnition implies t¯ij = tji . Moreover, tij is not symmetric in general. It is assumed that body force, body couple and energy supply can be decomposed into two parts; one is due to the electromagnetic ﬁelds, the other is supposed to be externally applied and known from the outset. Hence, ρFi = ρFie + ρFiext , ρLij = ρLeij ,

(2.2.8)

ρr = ρre + ρrext . Here, ρFie , ρLeij and ρre are thought to be expressed in terms of electromagnetic ﬁeld quantities, while ρFiext and ρrext are known. We have assumed that there are no externally applied body couples. For suﬃciently smooth ﬁelds, the global balance laws (2.2.1)–(2.2.4) assume their local forms

2.2 Equations of Balance

11

ρ˙ + ρx˙ i,i = 0 , ρ¨ xi = tij,j + ρFie + ρFiext , t[ij] = ρLe[ij] ,

(2.2.9)

ρU˙ = tij x˙ i,j − qi,i + ρre + ρrext , where use has been made of (2.2.8). Here, and throughout this work by x˙ i,j is meant −1 ˙ Fiα . (2.2.10) x˙ i,j = (x˙ i ),j = Fαj As follows from (2.2.9)1 , the present density ρ is related to ρ0 , the density in the reference conﬁguration, by ρ=

ρ0 . |J|

(2.2.11)

2.2.2 The Maxwell Equations As is well-known, there are several formulations of electrodynamics, all of which may be derived from diﬀerent postulations. For reference we refer the reader to Penfield and Haus ([177], Ch. 7). It is not our intention to derive the electromagnetic balance laws from various charge and electric circuit models, because we aim at describing the equivalence of these formulations rather than emphasizing their diﬀerences. We start with the Conservation of Magnetic Flux. Let Bi be the magnetic ﬂux density (or magnetic induction) and let Ei be the eﬀective electric ﬁeld strength, sometimes also called the electromotive intensity. The conservation law of magnetic ﬂux (Faraday law) may then be expressed as d Bi dai + Ei dli = 0 . (2.2.12) dt S

∂S

This relation must hold for any material surface S with boundary ∂S in R3 . By applying (2.2.12) to a closed surface S = ∂V, one can derive the Gauss–Faraday law, which states that   d   ⇒ Bi dai = 0 Bi dai = 0 , (2.2.13) dt ∂V

∂V

in which the constant of integration has been set to zero. For suﬃciently smooth ﬁelds, the balance laws (2.2.12) and (2.2.13) can be written in local form. To this end, we use the divergence, Stokes’ and Reynolds’ transport theorems as follows:

12

2 Basic Concepts

Bi dai

=

Bi,i dv , V

∂V

Ei dli ∂S

d dt

eijk Ei,j dai ,

=

(2.2.14)

S

Bi dai = S

B i dai . S

Substitute these into (2.2.12) and (2.2.13) and localize the emerging equations. This yields

eijk Ek,j + B i = 0, Bi,i = 0 .

(2.2.15)

In the above

eijk

 1, if i, j, k is an even pemutation of 1, 2, 3 ,     = −1, if i, j, k is an odd pemutation of 1, 2, 3 ,     0, else ,

(2.2.16)

` tensor and the convective is the completely skew-symmetric Levi–Civita

derivative ψ i of a vector ψi is deﬁned by

ψ i :=

∂ψi + ψi,j x˙ j + ψi x˙ j,j − ψj x˙ i,j . ∂t

(2.2.17)

¨ hnk A proof of (2.2.14)3 is given e.g. in Chadwick [39] or Hutter & Jo [100]. The second basic law of electromagnetism is the law of Conservation of Charge. Let Q be the electric charge density and Ji the conductive current. The conservation law of electric charges may then be expressed by the global balance law d Qdν + Ji dai = 0 . (2.2.18) dt V

∂V

This equation holds for any material volume V with boundary ∂V. It is a purely formal matter to introduce a ﬁeld Di such that Qdν = Di dai , (2.2.19) V

∂V

holds for any material part V of the body. This introduction of Di suggests to call it charge potential; however, we shall refer to Di as the dielectric displacement. It follows from (2.2.18) and (2.2.19) that the conservation of charge is satisﬁed if a ﬁeld Hi exists such that

2.2 Equations of Balance

Ji dai + S

d dt

13

Hi dli

Di dai = S

(2.2.20)

∂S

holds for any material surface S with boundary ∂S. Indeed, if we choose S = ∂V in (2.2.20), then ∂S = ∅ and (2.2.20) reduces to (2.2.18). Equation `re’s law. We shall call Hi the eﬀective magnetic ﬁeld (2.2.20) is called Ampe strength or the magnetomotive intensity. Note that Hi and Di as introduced by (2.2.19) and (2.2.20) are not unique. Again, assuming suﬃcient smoothness of the ﬁelds, the balance laws (2.2.18), (2.2.19) and (2.2.20) may be brought into local form. From (2.2.19) and (2.2.20) we then obtain Di,i = Q ,

eijk Hk,j − Di = Ji ,

(2.2.21)

∂Q ∂Q = Ji,i + =0, ∂t ∂t

(2.2.22)

and from (2.2.18) (Ji + Qx˙ i ),i +

which is the local conservation law of electric charges. For convenience we have also introduced here the non-conductive current Ji deﬁned by Ji := Ji + Qx˙ i .

(2.2.23)

The equations (2.2.21) are called the inhomogeneous Maxwell equations, in contrast to the homogeneous ones, (2.2.15), because on the right-hand sides they contain electric charge and current densities. Note that not all of the above equations are independent. Indeed, the conservation law of charge may be derived from the inhomogeneous Maxwell equations. In other words, when integrating the Maxwell equations, the conservation law of electric charges must be fulﬁlled along with the Maxwell equations. As a result, at most seven of the above electromagnetic ﬁeld variables can be considered basic, while for the remaining ones constitutive equations must be established. Of course, this does not mean that one cannot establish constitutive equations for more electromagnetic ﬁeld variables. Indeed, there is a valid point to treat both Ji and Q as dependent constitutive variables. If this is done, however, further integrability conditions must be satisﬁed. In the above presentation we have not made any appeal to a speciﬁc model, how the moving material body may contribute to the electromagnetic ﬁeld vectors Bi , Ei , Di and Hi . We shall not do this here either and refer to the pertinent literature (Penfield and Haus [177], Fano, Chu and Adler [73], Truesdell and Toupin [244], Pao [172], Eringen and Maugin [70] [71], Maugin et al. [137], Tiersten [238], Zhou [215, 216] and others). The reader may be puzzled, however, by the occurrence of the convective time derivatives in (2.2.15) and (2.2.21), which contain implicitly the motion. If

14

2 Basic Concepts

he so desires, he may formally eliminate the latter by simply deﬁning two new ﬁelds through Ei = Ei + eijk x˙ j Bk ,

Hi = Hi − eijk x˙ j Dk .

(2.2.24)

Here, Ei and Hi are the well-known Minkowskian electric and magnetic ﬁeld strengths. We note that relations (2.2.24) are not the only ones through which the motion can be eliminated. We might further mention that of the four ﬁelds Bi , Ei , Di and Hi only two are considered basic, while the remaining two will have to be described by constitutive assumptions. In practice, some of the above ﬁeld quantities are replaced by others, which may allow a better physical insight, but this does not change the fundamental fact that there are two basic vectorial ﬁeld quantities, while the remaining ones must be expressed in terms of the former. We shall come back to this at a later stage. 2.2.3 Material Description The balance laws listed in the previous two sections are written in the spatial or Eulerian formulation. Corresponding to these equations there is a material or Lagrangean description. Let dν, dak and dlk be the volume-, area- and length-increments in the spatial description, while dV, dAα and dLα are the corresponding increments in the material description. Then the following well-known identities hold between these increments: dν = |J|dV ,

−1 dak = JFαk dAα ,

dlk = Fkα dLα .

(2.2.25)

Here a remark concerning the second equation is in order. Since the normal vector on a closed surface is deﬁned as the outward normal and outward normals given by relation (2.2.25)2 are transformed into inward normals when sgnJ < 0, we must replace in (2.2.25)2 J by |J| whenever the surface is closed. Hence, if dAα and dak are area-elements of a closed surface, we have −1 dak = |J|Fαk dAα .

(2.2.26)

Although we may restrict J to positive values, use of |J| instead of J in the following equations is preferred in order to make the respective quantities objective under the full group of (Euclidian) transformations, (see Sect. 1.6). With the relations (2.2.25) the mechanical balance laws (2.2.1)–(2.2.4) may be transformed into the following forms d ρ0 dV = 0 , (2.2.27) dt VR

2.2 Equations of Balance

d dt d dt

ρ0 x˙ i dV = VR

∂VR

1 ρx˙ i x˙ i + ρU 2

Tiα dAα +

(2.2.28)

ρ0 (Lij + x[i Fj] )dV ,

(2.2.29)

VR

(x˙ i Tiα − Qα )dAα +

dV =

VR

ρ0 Fi dV ,

VR

x[i Tj]α dAα +

15

∂VR

ρ0 x[i x˙ j] dV = VR

d dt

(ρ0 x˙ i Fi + ρ0 r)dV .

VR

∂VR

(2.2.30) Here, integration is over reference volume and surface, respectively. Further, Tiα is the ﬁrst Piola–Kirchhoff stress tensor and Qα is the material energy ﬂux vector, which are related to tij and qi according to −1 tij , Tiα = |J|Fαj

tij = |J −1 |Fjα Tiα ,

−1 Qα = |J|Fαi qi ,

qi = |J −1 |Fiα Qα .

(2.2.31)

Assuming for the external source terms decompositions similar to those listed in (2.2.8) and supposing suﬃcient smoothness of the ﬁelds involved, we ﬁnd that (2.2.28)–(2.2.30) imply ¨i = Tiα,α + ρ0 Fie + ρ0 Fiext , ρ0 x T[iα Fj]α = ρ0 Lij ,

(2.2.32)

ρ0 U˙ = Tiα F˙ iα − Qα,α + ρ0 re + ρ0 rext , whereas (2.2.27) integrates to yield ρ0 = ρ0 (X). Apart from the stress tensors tij and Tiα we shall occasionally also use the so-called second Piola-Kirchhoff stress tensor deﬁned by −1 −1 −1 P Tαβ := Tiα Fβi = |J|tij Fαj Fβi .

(2.2.33)

P is symmetric if and only if tij is symmetric. There exists also Notice that Tαβ a material formulation of the electromagnetic balance laws (see [92]). To derive the material counterpart of (2.2.12), (2.2.13) and (2.2.18)–(2.2.20), recall that the integrals occurring in these equations must only be transformed back to the reference conﬁguration. With the aid of (2.2.25) we obtain straightforwardly d Bα dAα + Eα dLα = 0 , (2.2.34) dt SR

∂SR

Bα dAα = 0 , ∂VR

(2.2.35)

16

2 Basic Concepts

Dα dAα = −

d dt

∂VR

Dα dAα + SR

d dt

QdV , VR

QdV +

Jα dAα ,

(2.2.37)

SR

VR

Hα dLα =

∂SR

(2.2.36)

Jα dAα = 0 .

(2.2.38)

∂VR

Here, hollow quantities are the material counterparts of Bi , Ei , Di , Hi , Ji and Q and they are related to these according to the transformation rules Q = Q|J| ,

Q = |J −1 |Q ,

−1 Jα = Ji |J|Fαi ,

Ji = |J −1 |Fiα Jα ,

−1 Dα = Di |J|Fαi ,

Di = |J −1 |Fiα Dα ,

Hα = Hi Fiα sgnJ ,

−1 Hi = Fαi Hα sgnJ ,

−1 Bα = Bi JFαi ,

Bi = J −1 Fiα Bα ,

Eα = Ei Fiα ,

−1 Ei = Fαi Eα .

(2.2.39)

The quantities Eα , Bα etc. will be called the material or Lagrangean electromagnetic ﬁelds. Again, although J may be assumed positive we have written these formulas such that the hollow quantities transform under the Euclidian transformation group as scalars (see Sect. 1.6). For suﬃciently smooth ﬁelds the global laws (2.2.34)–(2.2.38) can be written in local form. Then they are Bα,α = 0 , eαβγ Eγ,β + B˙ α = 0 , eαβγ Hγ,β − D˙ α = Jα ,

Dα,α = Q ,

(2.2.40)

˙ + Jα,α = 0 , Q where all diﬀerentiations are with respect to the material coordinates. As was mentioned already before, this set of equations is a dependent one since the conservation of charge is already implicitly contained in the Maxwell equations. As a consequence, at most seven variables are independent, while the remaining ones must be given by constitutive equations. At the moment, the Lagrangean ﬁelds are introduced purely formally. That they are of advantage will be demonstrated in Chaps. 4 and 5.

2.3 The Entropy Production Inequality It is a fact of experience that real physical processes are irreversible. This means that processes cannot, in general, be traversed backward in time, i.e.,

2.3 The Entropy Production Inequality

17

time reversal does not lead to a physically realizable process. This fact is called the second law of thermodynamics and its mathematical realization is the entropy production inequality. It is based on the assumption that there exists an additive quantity η, called the entropy, which satisﬁes the balance law d ρηdν + φi dai − ρsdν = ργdν . (2.3.1) dt V

V

∂V

V

Here, φi is the entropy ﬂux, s the entropy supply and γ the entropy production. For suﬃciently smooth ﬁelds (2.3.1) implies ρη˙ + φi,i − ρs = ργ ,

(2.3.2)

and it is the expression of the second law of thermodynamics that γ≥0;

(2.3.3)

ρη˙ + φi,i − ρs ≥ 0 .

(2.3.4)

hence This inequality must hold for all thermodynamic processes, i.e. processes that satisfy the balance laws of thermomechanics and electrodynamics as well as the constitutive relations. We shall set the entropy supply s equal to the external energy supply divided by the absolute temperature Θ, i.e. s=

rext , Θ

(2.3.5)

but we shall, in general, not assume that “entropy ﬂux equals heat ﬂux divided by absolute temperature”. Thus, (2.3.4) becomes ρη˙ + φi,i ≥

ρrext . Θ

(2.3.6)

The material counterpart of (2.3.1) can easily be derived. In its local form, it reads ρ0 rext , (2.3.7) ρ0 η˙ + Φα,α ≥ Θ where −1 Φα = |J|Fαi φi , (2.3.8) is the Lagrangean entropy ﬂux. Before we proceed it seems to be worthwhile to justify the approach we take regarding the entropy inequality (2.3.4) and the interpretation we give to the entropy supply and to the entropy ﬂux. Our aim is not the justiﬁcation of one particular theory against any other one. On the contrary, we shall adopt certain results obtained by using a particular entropy principle in each theory and aim at a comparison of such theories. This comparison should be made on the level of fully developed theories.

18

2 Basic Concepts

2.4 Jump Conditions Let ˆ Σ = Σ(X, t) = σ ˜ (x, t) = 0 ,

(2.4.1)

be a smooth orientable surface, not necessarily material, and let WN and wn be its speed of propagation and speed of displacement, respectively, i.e. WN := −

ˆ ∂ Σ/∂t , ˆ α )1/2 ˆ α Σ, (Σ,

wn := −

∂σ ˜ /∂t . (˜ σ ,i σ ˜ ,i )1/2

(2.4.2)

Assume further that the electromagnetic ﬁeld quantities Bi , Ei etc., or Bα , Eα etc., as well as all mechanical quantities listed in Sect. 1.3.1 or 1.3.3 may suﬀer ﬁnite jumps across the surface Σ. In particular we assume χi (X, t) to be continuous on Σ, but its ﬁrst derivatives may suﬀer ﬁnite jumps. Hence, it follows that, although ˆ Σ(X, t) = σ ˜ (χ(X, t), t) = σ ˆ (X, t) , is one and only one surface in VR , the values for WN are the same on both sides of the surface only when χ˙ i and Fiα are continuous. Waves in which χ˙ i and Fiα may jump are called shock waves. By applying the global balance laws to a part of the body (V or VR ) containing the surface of discontinuity, we can derive jump conditions for the ﬁelds occurring in these laws. We suppose the methods of derivation to be known and therefore only list the results. The jump conditions obtained thereby will depend on whether one is dealing with the material or the spatial description. In the spatial description, the balance laws (2.2.12), (2.2.13) and (2.2.18)–(2.2.20) reveal that eijk Ej nk + x˙ i Bk nk − Bi (x˙ k nk − wn ) = 0 , Bi ni = 0 , Di ni = 0 , eijk Hj nk + x˙ i Dk nk − Di (x˙ k nk − wn ) = 0 , (2.4.3) Ji ni − Qwn = 0 . Here,

Φ := Φ+ − Φ− ,

(2.4.4)

denotes the jump of Φ across the surface σ ˜ (x, t), Φ± are the values of Φ immediately on the positive and negative side of the surface, respectively, and n is the unit normal vector at a point on the singular surface pointing into the positive side of σ ˜ (x, t). We remark that conditions (2.4.3) tacitly assume that Ji and Q are ﬁnite on the surface of discontinuity. This is a restriction; e.g. it excludes surface distributions of charge and current.

2.4 Jump Conditions

19

In the material description the counterparts of (2.4.3) read as follows Bα Nα = 0 , eαβγ Eβ Nγ + Bα WN = 0 , Dα Nα = 0 , eαβγ Hβ Nγ + Dα WN = 0 , (2.4.5) Jα Nα − QWN = 0 . If we set WN = 0, or wn = x˙ + ˙− i ni = x i ni , then the surface of discontinuity is material. In that case, the jump conditions may serve as boundary conditions. We like to note here that, in the sense of the deﬁnition given above, the boundary of a body in a vacuum is not a material surface of discontinuity because on this boundary: x˙ + i =0,

x˙ − i ni wn .

and

(2.4.6)

For the derivation of the jump conditions of momentum and energy, we start with the expressions for the electromagnetic body force and energy supply. We consider it to be known that the electromagnetic momentum and energy supply terms appearing in (2.2.2), (2.2.4), (2.2.28) and (2.2.30) can always be written in the form ρFie = tM ij,j + e

ρr +

ρFie x˙ i

∂gi , ∂t

M ρ0 Fie = Tiα,α + G˙ i ,

(2.4.7)

∂ω , = πi,i + ∂t

e

ρ0 r +

ρ0 Fie x˙ i

= Πα,α + Ω˙ .

M In the above relations, Tiα and tM ij are electromagnetic stress tensors, which sometimes are called Maxwell stress tensors, Gi and gi are the electromagnetic momenta in the material and spatial description, respectively, Πα and πi are the material and spatial representation of the electromagnetic energy ﬂux and Ω and ω are electromagnetic energy densities. All these quantities are expressible in terms of the electromagnetic ﬁelds, the motion and their derivatives. Hence, although we assume the electromagnetic ﬁelds to suﬀer at most a ﬁnite jump across the singular surface Σ, such an assumption does not hold in general for ρFie and ρre (or ρ0 Fie and ρ0 re ), which may become unbounded because of the occurrence of the above mentioned gradients. Apparently, for the weak forms of (2.4.7) one must write d ρFie dν = gi dν + (tM ˙ j )daj , ij − gi x dt V

V

V

d (ρre + ρFie x˙ i )dν = dt

in the spatial formulation, and

∂V

ωdν + V

(2.4.8)

∂V

(πi − ω x˙ i )dai ,

20

2 Basic Concepts

ρ0 Fie dV VR

d = dt

VR

e

(ρ0 r +

M Tiα dAα ,

Gi dV +

ρ0 Fie x˙ i )dV

∂VR

d = dt

VR

ΩdV + VR

(2.4.9)

Πα dAα ,

∂VR

M M in the material description, respectively. The ﬁelds Gi , gi , Tiα , tij , Πα , πi , Ω and ω are related to each other by

Gi = |J|gi , −1 M M Tiα = |J|Fαj (tij − gi x˙ j ) , −1 Πα = |J|Fαi (πi − ω x˙ i ) ,

(2.4.10)

Ω = |J|ω . As long as all ﬁelds are suﬃciently smooth, the left-hand and right-hand sides of (2.4.8) and (2.4.9) are equivalent. This is no longer so if ρFie and ρre may be unbounded. Then the left-hand sides of (2.4.8) and (2.4.9) may not be meaningful at all, whereas the expressions on the right-hand sides still make sense. We therefore postulate the expressions on the right-hand sides to be the appropriate global statements. Substituting (2.2.8)1,3 into (2.2.2) and (2.2.4) and using (2.4.8) yields the global form of the balance laws of linear momentum and energy appropriate for the derivation of the jump conditions. Under the assumption that the external sources ρFiext and ρrext remain ﬁnite at Σ, these global laws, together with the balance of mass (2.2.1), lead to the following set of jump conditions ρ(x˙ i ni − wn ) = 0 , (ρx˙ i − gi )(x˙ j nj − wn ) − tij + tM ˙ j nj = 0 , ij − gi x 1 ( 2 ρx˙ i x˙ i + ρU − ω)(x˙ j nj − wn ) − x˙ i tij − qj + πj − ω x˙ j nj = 0 . (2.4.11) Similarly, in the material description, ρ0 WN = 0 , M (ρ0 x˙ i − Gi )WN − Tiα + Tiα Nα = 0 , (2.4.12) 1 ( 2 ρ0 x˙ i x˙ i + ρ0 U − Ω)WN − x˙ i Tiα − Qα + Πα Nα = 0 . Note that the balance of mass simply implies that ρ0 may jump only along with a jump of WN , or on material surfaces, where WN = 0. For WN = 0, (2.4.11)2,3 constitute the boundary conditions for the tractions and the energy ﬂux of matter and ﬁelds, respectively. On the other hand, if ρ0 is continuous on Σ, (2.4.12)1 implies continuity of WN .

2.5 Material Objectivity

21

2.5 Material Objectivity Although we consider the principle of material frame indiﬀerence to be known in general, we would like to call upon it here, mainly because of the complications resulting from the combination of the mechanical balance laws with those of electrodynamics. Let xi be the Cartesian coordinates of a particle as measured by a stationary observer and let xi be the Cartesian coordinates of the same particle as measured by another observer in his frame of reference. We call transformations that relate xi with xi by a rigid-body motion Euclidian transformations. They have the form xi = Oij (t)xj + bi (t) ,

t = t ,

(2.5.1)

where Oij is an orthogonal (time dependent) matrix and where bi is an arbitrary (time dependent) vector. A Euclidian transformation for which Oij is not time dependent and for which bi = −Vi t is called a Galilei transformation, and it is well-known that the balance laws of classical non-relativistic mechanics are invariant and frame independent under such transformations. They are, however, frame dependent with respect to Euclidian transformations. Let a, a and A be scalar, vector and second rank tensor quantities over R3 , and assume that two observers in their reference systems {O, e1 , e2 , e3 }, {O , e1 , e2 , e3 } measure the components a, ai , Aij

and

a , ai Aij ,

(2.5.2)

respectively. Then the starred and the unstarred quantities are related to one another through a transformation that is dictated by the transformation group from the unstarred to the starred reference system. If this transformation has the form a = (det O)p a , ai = (det O)p Oij aj ,

(2.5.3)

Aij = (det O)p Oik Oil Akl , with p = 0 or p = 1, then a, a, A are called an objective scalar, objective vector and objective tensor, respectively. For p = 0 (p = 1) these quantities are speciﬁed as polar (axial) objective scalar, vector or tensor, respectively. For brevity, the attribute “polar” is often omitted for polar ﬁeld quantities. For instance, det O is an axial scalar, the curl is an axial vector operator, but the gradient is polar. If ai = x˙ i is the velocity it is easily seen that under the Galilei group it is an objective polar vector, but when the group is Euclidian the velocity is not an objective vector quantity. In contrast to classical mechanics, the Maxwell equations are neither invariant under Euclidian nor under Galilei transformations. The transformation group here is the extended Lorentz group. This group may be

22

2 Basic Concepts

explained as follows. Let (xi , t) denote the position xi and time t of a particle as measured by one observer and let (xi , t ) be those as measured by another observer, who translates relative to the ﬁrst observer with constant velocity Vi . A transformation of the form x V 1 t k k − 1 − V2 1 − V 2 /c2 1 − V 2 /c2 2 2 = (xi − Vi t)(1 + O(V /c )) ,

xi = xi + Vi

(2.5.4)

xk Vk t− 2 1 c t = = t − 2 xk Vk (1 + O(V 2 /c2 )) , c 1 − V 2 /c2 is then called a special Lorentz transformation. Special in this transformation is that the frames xi and xi are parallel. If they are also rotated with respect to each other then xi on the left-hand side of (2.5.4)1 must be replaced by Oji xj , where Oij is a constant orthogonal matrix. The group of these transformations is called the extended Lorentz group. In four-dimensional notation (A, B = 1, 2, 3, 4) , (2.5.5) xA = ∗AB xB , where the four vector x(4) is the ordered quadrupel (x(3) , t) and, to within terms of the order of (V 2 /c2 ), ∗AB is given by   O11 O12 O13 −O1k Vk    O O22 O23 −O2k Vk  21   . (2.5.6) ΛAB =     O31 O32 O33 −O3k Vk    2 2 2 −V1 /c −V2 /c −V3 /c 1 The Lorentz group is the analogon to the Galilei group and there is no immediate analogon to the Euclidian group. The principle of material frame indiﬀerence (material objectivity) states that the material response ought not be frame-dependent. But with respect to which transformation group? Obviously, since classical mechanics is invariant under the Galilei group and the special theory of relativity is so under the Lorentz group, the material response must be invariant with respect to one of these groups, depending on whether one deals with classical mechanics or with special relativity. We consider it to be known that the principle of material objectivity of classical mechanics as stated by Noll requests the material response to be invariant under Euclidian transformations. There ¨ller have been objections raised against the general truth of this (see Mu [159]), but this will be of no relevance here, because the results of this monograph (except Chap. 8) will also hold true if we restrict ourselves in those cases to Galilei transformations.

2.5 Material Objectivity

23

In order to gain some insight into the transformation properties of the electromagnetic ﬁeld variables, recall that the ﬁelds Ei , Bi , Hi and Di introduced in (2.2.24) can be written in the form of two skew-symmetric covariant and contravariant four-tensors, namely as     0 B3 −B2 E1 0 H3 −H2 −D1     −B 0 −H B1 E2  0 H1 −D2  3 3      , η AB =   , ϕAB =       B2 −B1 0 E3   H2 −H1 0 −D3      −E1 −E2 −E3 0 D1 D2 D3 0 (2.5.7) and that these tensors transform under general transformations of the form xA (xB ) according to ϕAB =

∂xC ∂xD ϕCD , ∂xA ∂xB

η AB =

∂xA ∂xB CD η . ∂xC ∂xD

(2.5.8)

With σ A := (Ji + Qx˙ i , Q) ,

(2.5.9)

it is then a routine matter to show that the Maxwell equations (2.2.15) and (2.2.21) are given by eABCD where

eABCD

∂ϕCD =0, ∂xB

∂η AB = σA , ∂xB

 −1, if ABCD is an even permutation of 1234 ,     1, if ABCD is an odd permutation of 1234 , =     0, else ,

(2.5.10)

(2.5.11)

is the four-dimensional permutation tensor. The equations (2.5.8) hold for any transformation xA = xA (xB ). If one chooses Euclidian transformations, one can show that Q

transforms as an objective scalar ,

Ei , Di , Ji

transform as objective polar vectors ,

Hi , Bi

transform as objective axial vectors .

(2.5.12)

For the proof of these statements, the reader may consult Appendix A. In particular, these quantities are objective under Galilei transformations. In the sequel we shall also introduce other electromagnetic variables, such as polarization and magnetization vectors, some of which are also objective vectors in the above sense.

24

2 Basic Concepts

The Maxwell equations (2.5.10) would formally be invariant under general transformations of the form xA = xA (xB ), as can easily be checked by substituting (2.5.8) into (2.5.10), if there would not be a relation of the form η AB = η AB (ϕCD ) ,

(2.5.13)

that is not invariant under the most general transformations. Indeed, a relation (2.5.13) even exists in vacuo, in which case it reads Di = ε0 Ei ,

Hi =

1 Bi . µ0

(2.5.14)

These relations are sometimes referred to as the Maxwell–Lorentz aether relations. Equations (2.5.14) restrict the class for which the Maxwell equations (2.5.10) are invariant to the extended Lorentz group. In general, (2.5.13) gives a model for electromagnetic ﬁeld interactions with matter. Based on the properties (2.5.12) we may then request as is done classically, that the material response be invariant under the Euclidian group. This is an approximation, because the Maxwell equations can never be rendered Lorentz invariant this way. On the other hand, dependent on the choice of electromagnetic body force, body couple and energy supply, the balance laws of mechanics may be Galilei invariant this way. Theories of this nature have ¨ller [127], been proposed by, amongst others, Toupin [241], Liu and Mu Pao and Hutter [171], Alblas [9], Van de Ven [249], Hutter [95], De Groot and Suttorp [53], Eringen and Maugin [70, 71], Tiesten [238], Maugin et. al. [137], and Shu-Ang Zhou [216]. For reasons that will become apparent shortly, we shall call these treatments non-relativistic theories. Of course, the above selection of the transformation group is not satisfactory, and it should be replaced by one treating the mechanical and electromagnetic equations alike. Such a description must necessarily be relativistic. Hence, the material response must be invariant under the extended Lorentz group. Considering motions only that are small relative to the velocity of light, we may then drop all terms containing a factor c−2 (this statement depends on the choice of units and in the way we state it here, SI-units are implied). In this way one arrives at Maxwell equations which are Lorentz invariant except for terms with a c−2 -factor. The same holds true for the constitutive relations and the mechanical equations. Theories obtained in this fashion may also be called non-relativistic, because we may look at their constitutive treatment in the light of Euclidian transformations, as we shall soon see. Finally there are formulations in which only the terms of order x˙ 2 /c2 are dropped, while terms with a c−2 -factor (but no (x˙ 2 /c2 )-factor) are kept. Such theories will be called semi-relativistic. In order to render these ideas a little more precise, consider a stationary frame of Cartesian coordinates and a particle moving with velocity x˙ i . An inertial frame, in which the particle is instantaneously at rest, is called the rest

2.5 Material Objectivity

25

frame. Electromagnetic variables as measured by an observer in this frame are called rest frame values and they are related to those in the stationary frame by a Lorentz transformation. Rest frame values will be denoted by a ◦

superimposed ring, viz. Ei etc. Of course, the Maxwell equations also hold in the rest frame (whereby all variables and operations carry the symbol ◦). Transforming these equations back to the original frame reveals the transfor◦

◦

mation rules for the variables Bi , Bi , Ei , Ei etc. To within the semi-relativistic approximation the transformation rules are ◦

Bi = Bi + O(x˙ 2 /c2 ) ,

Bi = Bi −

1 eijk x˙ j Ek , c2

◦

Ei = Ei + O(x˙ 2 /c2 ) , ◦

Di = Di + O(x˙ 2 /c2 ) ,

Di = Di +

1 eijk x˙ j Hk , c2

◦

(2.5.15)

Hi = Hi + O(x˙ 2 /c2 ) , ◦

Ji = Ji + O(x˙ 2 /c2 ) , ◦

Q= Q −

1 x˙ i Ji + O(x˙ 2 /c2 ) . c2

From these deﬁnitions as well as (2.2.24) it follows that Bi , Ei , Di etc. are the ﬁelds Bi , Ei , Di etc. as measured by an observer travelling with the particle. Furthermore, under rigid rotations they transform like scalars and vectors, and, as is shown in the theory of relativity (see Mφller, [158], p. 199), they form within the semi-relativistic approximation the ﬁrst three components of ◦

a set of four-vectors. Hence Bi , Ei , Di , Hi , Ji and Q are vectors and scalars, objective under Lorentz-transformations to within the semi-relativistic approximation. Mere inspection of (2.5.15) shows, however, that the variables Bi and Di are not objective under Galileian- or Euclidian transformations, because according to (2.5.12) Bi and Di are. But, in the non-relativistic limit, where c−2 -terms are dropped Di ≈ Di

and

Bi ≈ Bi ,

(2.5.16)

and Di and Bi are under Euclidian transformations an objective vector and an objective axial vector, respectively. Multiplying both sides of equation (2.5.15)2 with µ−1 0 shows that 1 1 Bi = Bi − ε0 eijk x˙ j Ek , µ0 µ0

(2.5.17)

so that the second term on the right-hand side is no longer of order c−2 . Hence, Bi /µ0 cannot be objective non-relativistically (although Bi is). The

26

2 Basic Concepts

objective quantity must be Bi /µ0 and indeed it can be shown (see Appendix A) that Bi /µ0 is objective under Euclidian transformations. A similar argument holds for Di , because 1 1 Di = Di + µ0 eijk x˙ j Hk . ε0 ε0

(2.5.18)

Here again, Di /ε0 is the quantity that is objective non-relativistically and not Di /ε0 , although Di is. Needless to say that Euclidian invariance of Di /ε0 also follows from the general relativistic transformation properties (see Appendix A). This is why we have called theories non-relativistic which use the principle of material objectivity under Euclidian transformations. Hence, we may conclude by stating that in the non-relativistic approximation (neglecting c−2 -terms) the requirement of invariance of the material response under Galilei transformations (a special Euclidian transformation) is equivalent to the requirement of Lorentz invariance . The diﬀerence is at most a philosophical one.

2.6 Constitutive Equations The balance laws (2.2.9), (2.2.15) and (2.2.21) express the common physical properties enjoyed by a material exhibiting electromagnetic mechanical interactions. They comprise a set of diﬀerential equations for many more variables than there are equations. So they are no ﬁeld equations from which the ﬁeld quantities could uniquely be determined. In order to become ﬁeld equations the balance laws (2.2.9), (2.2.15) and (2.2.21) must be complemented by constitutive equations. Of course, in this regard various degrees of complexity are possible. In PART I, we shall restrict ourselves to magnetizable and polarizable solids, which deform elastically under the action of electromagnetic and thermal ﬁelds and which exhibit electrical and thermal conduction. Mechanical dissipation is left out of consideration as is the exchange interaction and magnetic spin. In PART II, Chap. 8, viscous and plastic eﬀects will also be included. To obtain ﬁeld equations it must ﬁrst be decided which physical variables we suppose to be the independent ﬁelds. With regard to thermo-mechanical variables these ﬁelds are generally the motion X (X, t) and the temperature Θ(X, t). The basic electromagnetic ﬁeld variables are generally two electromagnetic ﬁeld vectors and the free charge. Any constitutive relation must be expressed therefore as a functional of the motion, the temperature, the free charge and two electromagnetic ﬁeld variables and derivatives thereof. Taking for instance Ei and Hi as the basic ﬁelds the conceivably simplest constitutive class exhibiting the above mentioned properties may have the form. ¯ i (t), x˙ i (t), Fiα (t), Ei (t), Hi (t), Θ(t), Θ,i (t), Q(t)) . C(t) = C(x

(2.6.1)

2.6 Constitutive Equations

27

Here, C stands for any scalar, vector or tensor valued quantity for which a ¯ constitutive equation is established, and C(·) is a function of the indicated variables. If the value of C at instant t depends on the motion xi , the temperature Θ and the electromagnetic ﬁelds Ei , Hi and Q at the same instant only, ¯ also we call the material response to have no memory. Now the function C(·) depends on x˙ i (t). Hence, with regard to xi the material appears to remember the past history of a process it has undergone for a very short time. Indeed knowing xi and x˙ i at a time t allows us to approximate the motion arbitrarily close to times τ = t as close as we please. We shall show, however, that an explicit dependence on x˙ i is not possible. As is usual in continuum mechanics, we require the constitutive relations to be independent of the observer. In terms of the above quantities C this principle of material objectivity requires that the quantities C do not only transform as objective scalars, vectors and tensors, respectively, but that ¯ C(·) is frame independent under the transformation group considered. In view of the fact that we shall restrict ourselves in the following chapters to a non-relativistic approach we require the constitutive equations to be frame independent scalar, vector and tensor valued functions under the group of Euclidian transformations. (Actually invariance under Galilei transformations is suﬃcient for constitutive relations of the form (2.6.1)). Then it is straightforward to shown that the constitutive quantities C cannot depend on xi and x˙ i explicitly. Furthermore, we have seen in the preceding section that Ei , Hi and Q are objective, and thus we may set ¯ iα , Ei , Hi , Θ, Θ,i , Q) . C = C(F

(2.6.2)

If viscous or plastic eﬀects are included then the adequate constitutive postulate is ¯ iα , F˙iα , Ei , Hi , Θ, Θ,i , Q) . C = C(F (2.6.3) Of course, since Di and Bi are also objective, it is always possible to replace in (2.6.2) Ei by Di and/or Hi by Bi . Moreover, one can formally introduce polarization Pi and magnetization Mi by Pi := Di − ε0 Ei ,

and

µ0 Mi := Bi − µ0 Hi ,

(2.6.4)

and then, assuming for the moment that Pi and µ0 Mi are an objective vector and an objective axial vector under Euclidian transformations, respectively, we could also use Pi and µ0 Mi as basic (i.e. independent) ﬁeld variables. In this way we can choose from nine possibilities for the constitutive relations of the kind (2.6.2). However, we shall not discuss them all here, but list below only those combinations, which are physically relevant and will appear in the next chapter. These are

28

2 Basic Concepts

ˆ iα , Pi , µ0 Mi , Θ, Θ,i , Q) , C = C(F ρ ρ M µ ˜ iα , Ei , 0 i , Θ, Θ,i , Q) , C = C(F ρ ¯ iα , Ei , Hi , Θ, Θ,i , Q) , C = C(F

(a) (b) (2.6.5)

(c)

Pi , Bi , Θ, Θ,i Q) , ρ ˇ iα , Ei , Bi , Θ, Θ,i , Q) . C = C(F +

C = C (Fiα ,

(d) (e)

Although all the arguments appearing in the constitutive relations (2.6.5) are objective quantities, we have not satisﬁed yet the requirement that + ˇ must be objective tensorial, vectorial and scalar ˆ ˜ ¯ C(·), C(·), C(·), C (·) and C(·) functions of their variables with respect to the Euclidian group. ˆ etc. is a The explicit form of these expressions depends on whether C(·) scalar, vector or tensor valued function. The representations in all these cases are well-known and are due to Noll. The reader may consult Truesdell and Noll’s treatise [243] for an account on the history and for a proof. If C is an objective scalar under the Euclidian transformation group it may be shown that its constitutive function is frame independent if ˆ αβ , Pα , Mα , Θ, Θ,α , Q) , C = C(C

(a)

˜ αβ , Eα , Mα , Θ, Θ,α , Q) , C = C(C

(b)

¯ αβ , Eα , Hα , Θ, Θ,α , Q) , C = C(C

(c)

+

C = C (Cαβ , Pα , Bα , Θ, Θ,α , Q) ,

(d)

ˇ αβ , Eα , Bα , Θ, Θ,α , Q) . C = C(C

(e)

(2.6.6)

Here, Cαβ is the right Cauchy-Green deformation tensor, deﬁned by (2.1.10)1 , and Pα := Eα := Ei Fiα ,

1 Pi Fiα , ρ

µ0 Mi Fiα sgnJ , ρ 1 Hα := Hi Fiα sgnJ , Bα := Bi Fiα sgnJ . µ0 Mα :=

(2.6.7)

If Ci is an objective polar vector, then ˆ γ (Cαβ , Pα , Mα , Θ, Θ,α , Q), etc. Ci = Fiγ C

(a)

(2.6.8)

(a)

(2.6.9)

If Ci is an objective axial vector, then ˆ γ (Cαβ , Pα , Mα , Θ, Θ,α , Q)sgnJ, etc. Ci = Fiγ C Finally, for a second-order tensor Cij one obtains

2.6 Constitutive Equations

ˆ γδ (Cαβ , Pα , Mα , Θ, Θ,α , Q), etc. Cij = Fiγ Fjδ C

(a)

29

(2.6.10)

We now shall outline the procedure that will be used to bring the constitutive equations in their ultimate form. First, we substitute the constitutive relations (2.6.5) for the dependent variables into the Maxwell equations and into the balance laws of mass, linear and angular momentum and energy. The resulting equations are the ﬁeld equations for ρ, χi , Θ, Q and the two basic electromagnetic ﬁelds (e.g. Pi and Mi in case a)). Any solution of these ﬁeld equations is called a thermodynamic process. Following Coleman and Noll, ¨ller [160], we request the entropy inequality, or any [44] or more generally Mu inequality derived from it, to hold for all smooth thermodynamic processes. This implies that at a particle we may freely choose the independent variables and derivatives thereof as long as the ﬁeld equations are not violated thereby. In an open system, that is for a body with arbitrary external body force ρFiext and energy supply ρrext no contributions are provided by the momentum and energy equations, because to any process there exist appropriate force and energy supply terms such that the momentum and energy equations are satisﬁed identically. Special care should be observed with regard to the electromagnetic variables, however, since, as can easily be deduced from the Maxwell equations, not all the gradients of the basic electromagnetic ﬁelds are arbitrary. Such gradients may occur in the entropy inequality, and if they do, the relations implied by the Maxwell equations must be fulﬁlled along with the exploitation of the entropy inequality. A detailed explanation of this point is given by Hutter [96]. In any case, once the constitutive relations are substituted into the entropy inequality and all the above mentioned side conditions are properly taken into account, an inequality results with terms that are explicitly linear in variables that, in a thermodynamic process, may have any arbitrarily assigned value. Therefore, since otherwise this inequality would be violated, each of the coeﬃcients of these variables must be identically zero. These conditions then imply the constitutive equations in their ultimate form. This procedure, which we assume to be familiar to the reader, will be applied (and described in greater detail) in the next chapter.

Part I

Equivalence of Diﬀerent Electromagnetic Formulations in Thermoelastic Solids

3 A Survey of Electromagneto-Mechanical Interaction Models

3.1 Preview The subject of electrodynamics of moving media has always been a controversial one. This book will not end or resolve all controversies, because we can answer some, but not all the relevant questions in connection with a complete thermodynamic theory of electromagnetism. The basic diﬃculties in the description of electromechanical interaction models are manifold. A ﬁrst diﬃculty is concerned with the invariance properties of the electromagnetic ﬁeld equations. As is well-known, Maxwell’s equations are invariant under Lorentz transformations, while the balance laws of classical mechanics are invariant under Euclidian transformations and frame indiﬀerent under Galilei transformations. Clearly, a proper derivation should also treat the mechanical equations relativistically. This is true, but for most problems of technical relevance, relativistic eﬀects are negligible. It is therefore customary, in general, to treat the mechanical equations classically, while the equations of electrodynamics are handled relativistically. In so doing it might in these theories become uncertain what transformation properties some variables are based upon. However, knowledge of such transformation properties is important, because they give us indications as to what variables are comparable among diﬀerent theories. A second and even more serious diﬃculty can be found in the deﬁnitions of electromagnetic body force, body couple and energy supply. The roots of this diﬃculty lie in the separation of the electromagnetic ﬁeld quantities in near and far ﬁeld eﬀects. This separation has been and still is the root of controversies, because almost every author separates the total ﬁelds diﬀerently. In other words, near and far ﬁelds are not unique. A third diﬃculty is connected with the Maxwell equations, which for deformable moving matter were ﬁrst derived by Minkowski. Apart from the Maxwell equations in Minkowski’s form there exists a variety of other forms of the Maxwell equations in deformable media, all of which are motivated from particular models. The “action” of the electromagnetic ﬁelds upon the material is described hereby by quantities referred to as polarization and magnetization. However, dependent upon the model of derivation, polarization and magnetization of one theory may be and in general are different from polarization and magnetization of another theory. Hence, while all K. Hutter et al.: Electromagnetic Field Matter Interaction in Thermoelastic Solids and Viscous Fluids, Lect. Notes Phys. 710, 33–88 (2006) c Springer-Verlag Berlin Heidelberg 2006 DOI 10.1007/3-540-37240-7 3

34

3 A Survey of Electromagneto-Mechanical Interaction Models

formulations of electromagnetism of deformable continua are equally valid – we know of ﬁve diﬀerent descriptions – special care must be observed that variables of one theory are not confused with those of another. As a ﬁnal diﬃculty, we mention that the equations of a theory of electromechanical interaction are highly nonlinear. Generally, they defy any exact analysis even for the most simple problems that are of physical relevance. As a result, linearization procedures are needed. To render the above statements more precise, consider the equations of motion which may be derived by formulating the balance law of momentum to an arbitrary part of the body. The local form of this balance law states that “mass times acceleration equals divergence of stress plus body force”. In ordinary classical mechanics the body force is either set equal to zero, or else given by the gravitational force. A body couple hardly occurs in applications, in which case the balance law of moment of momentum implies the symmetry of the (Cauchy) stress tensor. When the body under consideration is interacting with electromagnetic ﬁelds, however, body force and body couple are given by electromagnetic quantities. The total force and the total moment exerted on a body by electromagnetic ﬁelds may be separated into a long range and a short range eﬀect. The long range eﬀect is expressed as a body force and body couple. The short range eﬀect, on the other hand, manifests itself as surface tractions, which can be combined with the mechanical tractions giving rise thereby to the deﬁnition of the stress tensor. This decomposition is not unique, thus leading to non-unique body force expressions and non-unique stress tensors. As a consequence, the electromagnetic body couple cannot be unique either. Although this non-uniqueness might be quite striking to the novel reader it is nonetheless not disturbing at all if looked upon from the right point of view. Indeed, it is not important that the above separation into force and stress is unique, because diﬀerences in the body forces can always be absorbed in the stress tensors, provided that they are expressible as a divergence of a stress. A variety of mutually incompatible formulas for the force expressible in terms of stresses are therefore equivalent with respect to the total force. Only this force is physically observable. Thus the incompatibility is not physical, but metaphysical or semantic. The incompatibility expressed above also occurs in the energy equation (ﬁrst law of thermodynamics). This equation states that the time rate of change of the internal energy is balanced by the power of working of the stresses, the divergence of the heat ﬂux, and the energy supply due to electromagnetic eﬀects and due to heat. Since stress was already said to be nonunique, it follows that internal energy, heat ﬂux and electromagnetic energy supply cannot be determined uniquely either. Likewise, the electromagnetic energy supply might contain a term that is the divergence of a vector, which could be absorbed in the heat ﬂux vector. As an immediate consequence, it cannot be assured that heat ﬂux is energy ﬂux of thermal nature. We shall therefore prefer the term energy ﬂux instead.

3.1 Preview

35

As was the case for the momentum equation, seemingly incompatible expressions for internal energy, energy ﬂux and energy supply of electromagnetic origin do not prevent two theories from being equivalent. However, it is easily understandable that a proof of equivalence must be diﬃcult in general, for stress, internal energy and entropy (and also some electromagnetic ﬁeld vectors) are interrelated by thermodynamic conditions. More explicitly, thermodynamic requirements make stress and entropy (and other quantities) derivable from a so-called free energy. If two theories are diﬀerent in the body force, body couple and energy supply, therefore, the condition that the momentum and energy equations remain the same must amount to an interrelation between the free energies of two theories. Hence, equivalence of two theories of electromechanical interactions is a thermodynamic statement in general. The question of equivalence of two theories lies at the center of the diﬀerent formulations of electromechanical interaction theories. Although there is a valid point behind the statement that equivalence of diﬀerent theories need not be proved, because these theories describe diﬀerent physical situations, we nevertheless take the position that diﬀerent formulations of electromechanical interaction theories should yield the same results for physically measurable quantities, if the theories are claimed to be applicable to a certain class of material response. For instance, if we call a material a thermoelastic polarizable and magnetizable solid and if there is more than one formulation for such a solid, one should expect that, irrespective of all diﬀerences in the details, these theories will in any initial boundary value problem deliver the same results for physically measurable quantities. Measurable or observable quantities are all those which can be measured uniquely by two diﬀerent observers. All kinematical quantities that are derivable from the motion are measurable in principle and so is the (empirical) temperature. Regarding electromagnetic ﬁeld quantities, we take the position that they are not measurable except in vacuo where they can be observed by measuring the force on a test charge. There exist variables not observable by any means. These are all those which are not deﬁned except by the mathematical properties laid down for them. To demonstrate the equivalence of the diﬀerent formulations of electromechanical interaction theories it is necessary to prove that physically measurable quantities in two formulations assume the same values in every point of the body for any initial boundary value problem. This does not only mean that the ﬁeld equations of one theory must be transformable into those of the other, but this condition also includes the boundary and initial conditions. One of the major goals of this monograph is to give an exposition of the existing theories of polarizable and magnetizable electrically and thermally conducting materials and to show in what sense they can be called equivalent. The reasons behind this non-uniqueness of electrodynamics in moving media are twofold. For one, the action of the body on the electromagnetic ﬁelds is generally described by adding to the ﬁeld variables occurring in vacuo

36

3 A Survey of Electromagneto-Mechanical Interaction Models

two other electromagnetic ﬁeld vectors. This addition is not unique and results in diﬀerent forms of the Maxwell equations. Second, even when we restrict ourselves to a particular form of the Maxwell equations, the electromagnetic forces, couples and energy supply terms need not be unique. More precisely, we mention that the two electromagnetic ﬁeld vectors describing the interaction of a ponderable body can be introduced for instance by postulating that every material point is equipped with a number of noninteracting electric and magnetic dipoles. These dipole moments then form the two additional electromagnetic ﬁeld vectors, which are called polarization and magnetization. When the calculations with these postulated dipoles are carried through consistently, a certain set of Maxwell equations (now called the Chu formulation) emerges. These equations are diﬀerent from those which follow from the postulation that magnetization is modeled as an electric circuit, which follows the motion of the material particle in question (statistical and Lorentz formulations). As far as electromagnetic body forces are concerned these are not even unique when one is restricting oneself to a particular interaction model. Indeed, in the Chu formulation we shall present two versions of body force expressions and we shall prove that the two are not distinguishable by any measurements. This proof will also be given for all other formulations. However, we shall not present the models as such, because they are amply treated in the pertinent literature. Although the proof of the equivalence of various theories of electrodynamics in deformable continua is a very important achievement, we want to state here clearly that we have performed this proof only on the level of nonrelativistic theories. The exact deﬁnitions of the term “non-relativistic” will be made precise in the respective chapters. It may suﬃce to mention that it essentially means that in MKSA-units terms containing a c−2 -factor are neglected. Here, c is the speed of light in vacuo. There exists a number of other theories of electromechanical interactions in which it is claimed that terms of order V 2 /c2 are neglected (V = velocity of the particle in the body) while those containing c−2 -factor are kept. We term such approximations “semi-relativistic”. Quasistatic theories (terms, containing a c−1 -factor are neglected) will not be treated here. It is a well-known fact that ﬂuids are best handled in the spatial description. It is also known that electrodynamics is usually only formulated in the spatial description. However, for a theory of solids it would be advantageous when all equations could be referred to the reference conﬁguration. This is indeed possible and it essentially amounts to the introduction of new electromagnetic ﬁeld variables as introduced in Sect. 2.2.3 of Chap. 2, see formulas (2.2.39). It turns out that these so-called Lagrangean ﬁeld variables are much more convenient to describe the theory of solids, because many thermodynamic formulas appear in a more condensed form this way (see e.g.

3.1 Preview

37

the Maxwell equations (2.2.40)). Another reason for the introduction of the material description is its advantage in the linearization of the governing equations. This linearization procedure is substantially easier when performed in the material rather than in the spatial description. This brings us naturally to the linearization procedure of the various theories. In principle, there are two alternatives open to extract some useful information from these complicated equations. One is to ﬁnd numerical solutions for the nonlinear equations and the other is to linearize the equations on the basis of a sequence of consistent approximations. We shall follow the latter, because it provides a better access to the real physics of the problem. The linearization procedure is analogous to situations referred to as “small ﬁelds superimposed upon large ﬁelds”. The diﬀerence between these general treatments and ours is that the deformations are assumed to be small. This assumption is not necessary, and indeed the formal expansion procedures we shall apply also hold true for the general case. When the restriction to small deformations is used, however, it means physically that large external ﬁelds primarily induce strong electromagnetic ﬁelds within the body, but only small deformations. Therefore, in the ﬁrst step of evaluating the induced electromagnetism, the deformations may be neglected alltogether. A set of zerothorder equations, which formally agrees with rigid-body electrodynamics, is thus obtained. In the second step small strains are considered which add small but important corrections to the zeroth-order electromagnetic ﬁelds. Thus, the second set consists of linear ﬁeld equations, the coeﬃcients of which generally depend upon the zeroth-order electromagnetic ﬁelds. These ﬁeld equations may then be applied to solve problems such as magnetoelastic buckling, wave propagation in a material subject to electromagnetic ﬁelds, etc. Clearly, because we shall prove that all electromechanical interaction theories of polarizable and magnetizable solids are equivalent, the linearization procedure mentioned above need only be performed for one particular theory, which can be selected according to our needs. Moreover, it should be clear that this equivalence must amount in the statement how the free energy as a function of its independent variables in one theory is related to the free energy of another theory. The set of independent variables in this second free energy may very well be diﬀerent from the ﬁrst one. In other words, the correspondence relations for equivalence of various theories are dependent on which set of independent variables is chosen in the constitutive relations, but it is quite clear that the equivalence as such should not depend on the choice of the independent ﬁelds. From a theoretical point of view the problem just raised is not a serious one, because, in principle, equivalence of diﬀerent constitutive formulations in one single theory can be established quite easily. It then suﬃces to prove equivalence of two diﬀerent electromagnetic descriptions of deformable bodies with the aid of just one constitutive formulation in each of them.

38

3 A Survey of Electromagneto-Mechanical Interaction Models

To ﬁnd the free energy of a particular formulation from that of another one is a very diﬃcult problem in practice, however. It amounts to solving a functional diﬀerential equation the solutions to which are not known to date. Nevertheless, special cases are straightforward to handle. They serve as explicit examples, which should demonstrate that equivalence is possible. Mathematically this is important, because it serves as an explicit demonstration that the functional diﬀerential equations mentioned above do admit exact solutions. That these correspond to a reasonable physical situation is a nice additional property. The mathematical question of existence and non-existence of solutions will not be attacked here. Instead we look at the approximations in the way described below. It is customary in applications to write for the free energies polynomial expressions, and it is generally assumed that these polynomials can be truncated at a certain level. When polarization and magnetization are amongst the independent constitutive variables the free energy will be a polynomial expression in the deformation tensor, the temperature, polarization and magnetization, and the coeﬃcients in this polynomial expression give rise to eﬀects such as magnetic and electric anisotropy, magnetostriction, electrostriction etc. The coeﬃcients bear the names electric and magnetic susceptibilities etc. The same theory could be derived also with the electric ﬁeld strength and the magnetic induction as independent ﬁelds instead of magnetization and polarization. The free energy of this formulation would again be expressed as a polynomial of its variables and it would again be truncated at a certain level. This polynomial would again give rise to eﬀects such as electric and magnetic anisotropies, magnetostriction and electrostriction etc, but it is evident that the coeﬃcients of this polynomial must be diﬀerent from those of the other, if the two formulations aim at describing the same phenomena. The literature is full of confusion in this regard, mainly because diﬀerent coeﬃcients bear the same name. From the above it is, however, quite clear that there must be relations between the above mentioned coeﬃcients. We shall show how these relations look like and in what sense the emerging approximate theories can be regarded to be equivalent. The ﬁndings can be summarized as follows: Two formulations, in which the free energies are represented by polynomials of a certain order in their variables, can only be equivalent to within terms that were omitted in the expansion process. Only on the basis that these terms are negligibly small can we claim two theories to be equivalent. An analogous statement also holds for one single formulation in which certain constitutive quantities are interchanged as dependent and independent variables.

3.2 Scope of the Survey In this chapter we make an attempt to surveying various electromagnetic interaction models known to date. We do not present all the descriptions of magnetizable and polarizable deformable bodies, but list the ones that have

3.3 The Two–Dipole Models

39

received considerable attention in the recent literature only. A comparison of these models with still other ones will be given in Sect. 3.6. Any continuum theory of deformable bodies subject to electromagnetic ﬁelds amounts to the presentation of the basic electromagnetic ﬁeld variables, their relations to the other ﬁelds, as well as to the postulation of electromagnetic body force, body couple and energy supply. Once this is done, the Maxwell equations and the balance laws of mechanics and thermodynamics can be expressed in terms of the variables of the model in question. Using thermodynamic arguments, it is then a routine matter to reduce a postulated set of constitutive equations to a form compatible with the second law of thermodynamics. It is the purpose of this chapter to present the various models of electromechanical interactions, to scrutinize their invariance properties, to derive the constitutive theory in each peculiar case and to present each model such that, ﬁrstly, a comparison of one model with any other can be achieved fairly straightforwardly and, secondly, an initial boundary value problem can be solved, at least in principle.

3.3 The Two–Dipole Models As discussed by Fano, Chu and Adler [73], or Penfield and Haus [177], in the Chu formulation the Maxwell equations for moving matter are expressed in terms of Q, Ji , the velocity ﬁeld x˙ i , and four electromagnetic ﬁeld quantities EiC , HiC , PiC and MiC , which are related to the ﬁelds Di , Ei , Bi and Hi by the following transformation rules Di = ε0 EiC + PiC ,

Ei = EiC + µ0 eijk x˙ j HkC ,

Bi = µ0 HiC + µ0 MiC ,

Hi = HiC − ε0 eijk x˙ j EkC .

(3.3.1)

For reasons that will become apparent shortly we shall occasionally also make use of two auxiliary ﬁelds deﬁned by Bia := µ0 HiC ,

Dia := ε0 EiC .

(3.3.2)

As usual, EiC and HiC are the electric and magnetic ﬁeld strengths, and PiC and MiC are the polarization per unit volume and the magnetization per unit volume, respectively. In order to diﬀerentiate these ﬁelds from those occurring in other formulations, we have used a superscript C to indicate that these ﬁelds are those as deﬁned by Chu. Substituting (3.3.1) and (3.3.2) into the Maxwell equations (2.2.15) and (2.2.21), we obtain

40

3 A Survey of Electromagneto-Mechanical Interaction Models a C Bi,i = −µ0 Mi,i ,

∂MiC ∂Bia C = −µ0 + µ0 eijk (eklm x˙ l Mm ),j , ∂t ∂t C = Q − Pi,i ,

C eijk Ek,j + a Di,i

C eijk Hk,j −

(3.3.3)

∂Dia ∂PiC C = Ji + − eijk (eklm x˙ l Pm ),j . ∂t ∂t

In the above equations µ0 and ε0 are universal constants with ε0 µ0 = c−2 , c being the speed of light in vacuo. If we regard EiC and HiC as the basic electromagnetic ﬁelds, then, apart from Ji and Q, the terms on the right-hand sides of (3.3.3) may be interpreted as charge and current densities due to polarization and magnetization. By assuming that positive and negative electric and magnetic charges exist and may be combined to electric and magnetic dipole moments, the above charge and current densities due to polarization and magnetization can easily be derived, Pao & Hutter [171], Pao [172]. Equations (3.3.3) are therefore completely symmetric in the electric and magnetic ﬁeld quantities except that magnetic monopols are assumed not to exist, so that the corresponding charge and current densities are absent in (3.3.3). This interpretation is helpful for the derivation of electromagnetic body force, body couple and energy supply. The above set of electromagnetic variables is based on the postulations that: (i) only two vector quantities are necessary to describe the electromagnetic ﬁelds in free space, and (ii) material bodies contribute toward the electromagnetic ﬁelds by acting as sources for these ﬁelds. These sources are usually interpreted in terms of electric and magnetic dipoles, each such doublet being composed of a negative and positive electric and magnetic monopole. We would like to de-emphasize this interpretation, mainly because of the well-known objections physicists may rise against it. Nevertheless we accept the Chu formulation as a proper description, but view it as obtained from the deﬁnitions (3.3.1) by mere variable transformations. Before we proceed it is advantageous to investigate how the Chu-variables behave under the transformations discussed in Sect. 1.6. Following an approach analogous to that outlined by Truesdell and Toupin in [244], Sect. 283, it is not diﬃcult to show that under the Euclidian transformation group Q

transforms as an objective scalar ,

Ei , PiC , Ji

transform as objective vectors ,

Hi , µ0 MiC

transform as objective axial vectors .

It can also be shown that

(3.3.4)

3.3 The Two–Dipole Models

Dia

transforms as an objective vector, while

Bia

transforms as an objective axial vector ,

41

(3.3.5)

under the Euclidian transformation group. For the main lines of the proof the reader may consult Appendix A. It is clear that the above transformation rules also apply under the slightly less general Galilei group. The properties laid down in (3.3.4) and (3.3.5) are exact in the sense that the Maxwell equations can exactly (that is without the neglect of c−2 -terms) be rendered Euclidian invariant. However, relations (3.3.2) must also be satisﬁed, and they are not Euclidian invariant, but Lorentz invariant or Galilei invariant only to within terms containing a c−2 -factor (see also eqs. (3.3.6)7,8 ). Therefore, the Chu formulation of electromagnetism consisting of the Maxwell equations (3.3.3) together with the aether relations (3.3.2) is invariant under the Galilei group only within the non-relativistic approximation. Hence, in order to remain consistent, we must neglect in the following interaction models all terms preceded by a factor c−2 . In order to investigate semi-relativistic objectivity properties, let us consider the extended Lorentz group. Under these transformations the Chu variables transform (to within terms of order O(V 2 /c2 )) according to ◦

Q= Q −

◦

1 x˙ i Ji + O(V 2 /c2 ) , c2

◦

E i = Ei + O(V 2 /c2 ) , ◦

PiC = PiC + O(V 2 /c2 ) , ◦

Dia = Dia +

J i = Ji + O(V 2 /c2 ) ,

◦

Hi = Hi + O(V 2 /c2 ) , ◦

MiC = MiC + O(V 2 /c2 ) ,

(3.3.6)

1 eijk x˙ j HkC + O(V 2 /c2 ) , c2 ◦ 1 Bia = Bia − 2 eijk x˙ j EkC + O(V 2 /c2 ) . c

Thus it is obvious that the transformation rules (3.3.4) also hold under the extended Lorentz group, if terms of O(V 2 /c2 ) are dropped. There is only ◦

one exception, namely that Q has to be replaced by Q. In the non-relativistic approximation (c−2 = 0) this diﬀerence is negligible, however, and hence in ◦

this approximation we may everywhere replace Q by Q. Moreover, there are reasons to assume that Ji is of the order of QV . Then, Q may even ◦

be replaced by Q in the semi-relativistic approximation. As a consequence the Chu formulation of electromagnetism has the nice feature to obey the non-relativistic as well as the semi-relativistic invariance requirements. Note that, in contrast to (3.3.5), the auxiliary ﬁelds Dia and Bia do not behave as objective quantities under the Lorentz transformation group, not even in a semi-relativistic sense.

42

3 A Survey of Electromagneto-Mechanical Interaction Models

A complete theory of deformable, polarizable and magnetizable bodies consists of a set of electromagnetic and thermomechanical equations, the latter including expressions for the body force, body couple and energy supply due to the electromagnetic ﬁelds. On the non-relativistic level there are essentially two distinct formulations both of which use the dipole model not only for the polarization but also for the magnetization. 3.3.1 The Two–Dipole Model with a Nonsymmetric Stress Tensor (Model I) As mentioned above, to complete the description of deformable continua in the electromagnetic ﬁelds, expressions for ρFie , ρLeij and ρre are needed. Penfield and Haus [177] and Pao and Hutter [171] have motivated and derived the following expressions using the two-dipole model ρFie = QEiC + µ0 eijk Jj HkC C C + PjC Ei,j + µ0 eijk x˙ j Hk,l PlC

+

C µ0 MjC Hi,j

−

d + ρµ0 eijk dt

C ε0 eijk x˙ j Ek,l µ0 MlC

PjC ρ

d − ρε0 eijk dt

HkC µ0 MjC ρ

EkC ,

ρLeij = P[iC Ej] + µ0 M[iC Hj] , ρre = Ji Ei + ρEi

d PiC d µ0 MiC + ρHi . dt ρ dt ρ

(3.3.7) The complete derivation of (3.3.7) as given by Pao and Hutter [171], is based on the assumptions that: (i) each material particle is equipped with a number of mutually noninteracting electric and magnetic dipoles, (ii) each monopole suﬀers an electromagnetic body force as described by the Lorentz force (see below), and (iii) the monopoles of a particular dipole are only a small distance apart so that Taylor series expansions are justiﬁed. The Lorentz force mentioned in assumption (ii) is hereby taken in the form ρFiLorentz = Qe EiC + µ0 eijk Qe x˙ j HkC + Qm HiC − ε0 eijk Qm x˙ j EkC ,

(3.3.8)

where Qe and Qm are the electric and magnetic charges of the monopoles. Again we do not wish to emphasize the physical aspects of this approach, but we would like to view (3.3.7) as possible interaction postulates. We refer to the paper of Pao and Hutter [171], for the details of the derivation. Substituting (3.3.7) into the balance laws (2.2.9) yields the local balance laws of mass, linear and angular momentum and energy in the form

3.3 The Two–Dipole Models

ρ˙

43

+ ρx˙ i,i = 0 ,

ρ¨ xi = tij,j + ρFiext + QEiC + µ0 eijk Jj HkC d PjC C Hk dt ρ d µ0 MjC C Ek − ρε0 eijk dt ρ

C C +PjC Ei,j + µ0 eijk x˙ j Hk,l PlC + ρµ0 eijk C C +µ0 MjC Hi,j − ε0 eijk x˙ j Ek,l µ0 MlC

t[ij] = P[iC Ej] + µ0 M[iC Hj] , d PiC d µ0 MiC + ρHi + ρrext . dt ρ dt ρ (3.3.9) It is a routine though rather elaborate matter to transform (3.3.7) to the form listed in (2.4.7). One obtains ρU˙ = tij x˙ i,j − qi,i + Ji Ei + ρEi

= ε0 EiC EjC + µ0 HiC HjC + Ei PjC + µ0 Hi MjC 1 − δij ε0 EkC EkC + µ0 HkC HkC , 2 1 I gi = − 2 eijk EjC HkC , c I M tij

(3.3.10)

πi = −eijk EjC HkC + PiC EjC + µ0 MiC HjC x˙ j , 1 I ω = − ε0 EiC EiC + µ0 HiC HiC . 2 I

Here we have used a left upper index to distinguish these quantities from those of other interaction models. With the expressions (3.3.10) the jump conditions of momentum and energy as listed in (2.4.11) are readily derived. For the sake of easy reference they are given below together with the jump conditions of electromagnetic ﬁelds and mass µ0 HiC + µ0 MiC ni = 0 , ε0 EiC + PiC ni = 0 , eijk EjC nk + µ0 x˙ i MkC nk + µ0 HiC wn − µ0 MiC (x˙ k nk − wn ) = 0 , eijk HjC nk − x˙ i PkC nk − ε0 EiC wn + PiC (x˙ k nk − wn ) = 0 , ρ(x˙ i ni − wn ) = 0 , tij + ε0 EiC EjC + µ0 HiC HjC + Ei PjC + µ0 Hi MjC 1 − 21 δij ε0 EkC EkC + µ0 HkC HkC + 2 eikl EkC HlC x˙ j nj (3.3.11) c 1 − (ρx˙ i + 2 eikl EkC HlC )(x˙ j nj − wn ) = 0 , c

44

3 A Survey of Electromagneto-Mechanical Interaction Models

x˙ i tij − qj − eijkl EkC HlC + PijC EiC + µ0 MjC HiC x˙ i + 12 ε0 EkC EkC + µ0 HkC HkC x˙ j nj 1 1 C C C C ε ( x ˙ − ρ x ˙ x ˙ + ρU + E E + µ H H n − w ) =0. i i 0 0 j j n i i i i 2 2 Note that the jump conditions (3.3.11), specialized for material surfaces, have also been obtained by Pao and Hutter, [171]. Their jump conditions for the stress, however, does not contain the term 1 C C eikl Ek Hl x˙ j nj , c2 but, as we shall soon see, this term is negligible in a non-relativistic theory as it contains a prefactor c−2 . Before discussing the constitutive equations, it is interesting to look at the invariance properties of the electromagnetic body force, body couple and energy supply. At the beginning of this section the invariance properties of the electromagnetic ﬁeld variables were listed. From these it is immediately seen that Leij transforms under the Euclidian group as an objective skew-symmetric tensor. Under the extended Lorentz group Leij still transforms as an objective skew-symmetric tensor, but only to within terms of order O(V 2 /c2 ). Hence, Leij is objective non-relativistically as well as semirelativistically. If we write the energy supply term (3.3.7)3 in the form

ρre = (Ji + Pi )Ei + µ0 Mi Hi + (Ei Pj + Hi µ0 Mj )x˙ i,j ,

(3.3.12)

then we recognize that ρre is not an objective scalar under the Euclidian transformation group because of the term involving x˙ i,j . Incidentally, in (3.3.12) we have written Pi and µ0 Mi for PiC and µ0 MiC in order to stress the invariance properties of PiC and µ0 MiC . Moreover, it should be noted that the material time derivative of an objective vector is not an objective vector under the Euclidian group; but the convective time derivative is objective. This is the reason why we have tried to use convective time derivatives in (3.3.12). The non-objective part of (3.3.12) is now given by (Ei Pj + µ0 Hi Mj )x˙ [i,j] , which, in view of (3.3.9)3 , equals − t[ij] x˙ [i,j] . With use of this result, (3.3.12) shows that

ρre + t[ij] x˙ [i,j] = (Ji + Pi )Ei + µ0 Mi Hi + (Ei Pj + µ0 Hi Mj )x˙ (i,j) , (3.3.13)

3.3 The Two–Dipole Models

45

is an objective scalar. Hence, the balance of internal energy can be written as

ρU˙ = t(ij) x˙ (i,j) − qi,i + (Ji + Pi )Ei + µ0 Mi Hi + (Ei Pj + µ0 Hi Mj )x˙ (i,j) , (3.3.14) a form in which each term is an objective scalar under the Euclidian transformation group. In much the same way it can be shown that each term in the above equation is – to within terms of order O(V 2 /c2 ) – also an objective scalar under the extended Lorentz group. Hence, the energy balance law is an invariant equation in the non-relativistic and semi-relativistic sense. Finally, we also investigate the transformation properties of the electromagnetic body force (3.3.7)1 . A straightforward calculation shows that ρFie can be written in the following form

C ρFie = (Q − Pj,j )Ei + eijk (Jj + PjC )Bka C −µ0 Mj,j Hi

−

eijk µ0 MjC

Dka

+

(Ei PjC

(3.3.15) +

Hi µ0 MjC ),j

.

Using the transformation properties stated in (3.3.4) and (3.3.5), we readily see that each term on the right-hand side of this equation is an objective (polar) vector under the Euclidian transformation group. Because of the transformation properties of the auxiliary ﬁelds Dia and Bia , the body force does not enjoy the invariance properties of semi-relativistically correct Lorentz transformations, however. The above expression of the electromagnetic body force is written in a form which still contains the auxiliary ﬁelds. These ﬁelds may be eliminated with the aid of relations (3.3.2), which, however, are only invariant equations in the non-relativistic approximation. A straightforward calculation, in which the Maxwell equations (3.3.3) are used and in which terms of order O(V 2 /c2 ) are neglected, then shows that ρFie can be written in the form

ρFie = QEi + µ0 eijk Jj Hk + Pj Ei,j + µ0 eijk Pj Hk 1 (3.3.16) [(Jj + Pj )(x˙ i Ej − x˙ j Ei ) 2 c d +eikl Mj El x˙ k,j + ρeijk (Mj /ρ)EkC ] . dt In this form the body force is no longer an objective vector under Euclidian transformations; this objectivity is destroyed by use of relations (3.3.2) for the elimination of Dia and Bia . However, in a non-relativistic approximation the term involving a c−2 -term must be dropped. Hence, within a consistent nonrelativistic approximation the momentum equation (3.3.9)2 must be written as ρ¨ xi = tij,j + ρFiext + QEi + µ0 eijk Jj Hk + Pj Ei,j (3.3.17) +µ0 eijk Pj Hk + µ0 Mj Hi,j ,

+µ0 Mj Hi,j +

46

3 A Survey of Electromagneto-Mechanical Interaction Models

or since

QEi + µ0 eijk Jj Hk + Pj Ei,j + µ0 eijk Pj Hk + µ0 Mj Hi,j = {ε0 Ei Ej + µ0 Hi Hj + Ei Pj + µ0 Hi Mj

(3.3.18)

− 12 δij (ε0 Ek Ek + µ0 Hk Hk )},j + O(V 2 /c2 ) as ρ¨ xi = tij,j + ρFiext +{ε0 Ei Ej + µ0 Hi Hj + Ei Pj + µ0 Hi Mj − 12 δij (ε0 Ek Ek + µ0 Hk Hk )},j .

(3.3.19)

Moreover, the c−2 -terms in (3.3.10) and (3.3.11) must also be neglected so that I gi vanishes, i.e. I gi = 0 . (3.3.20) Thus the jump conditions for momentum and energy reduce to tij + ε0 Ei Ej + µ0 Hi Hj + Ei Pj + µ0 Hi Mj − 12 δij (ε0 Ek Ek + µ0 Hk Hk ) nj − ρx˙ i (x˙ j nj − wn ) = 0 , and

(3.3.21)

x˙ i tij − qj − ejkl Ek Hl + (ε0 Ej Ek + µ0 Hj Hk )x˙ k

+(Pj Ek + µ0 Mj Hk )x˙ k − 12 (ε0 Ek Ek + µ0 Hk Hk )x˙ j nj − { 12 ρx˙ i x˙ i + ρU + 12 (ε0 Ei Ei + µ0 Hi Hi )}(x˙ j nj − wn ) = 0 .

(3.3.22)

To summarize, we may state that the Maxwell equations together with the balance laws of mass, momentum (in the original not approximated version), angular momentum and energy presented here are Euclidian invariant. However, since these equations must be supplemented by the MaxwellLorentz aether relations (3.3.2), the complete set of electromagnetic and mechanical ﬁeld equations is Euclidian invariant only in the non-relativistic approximation. With this digression on invariance properties, which in our opinion are crucial, we now continue with the constitutive theory, which in accordance with the preceding conclusions must also be Euclidian invariant. The conditions for these requirements to be satisﬁed are given in Sect. 1.7, and in equation (2.6.5) some possible constitutive relations are presented. In the Chu formulation the B-ﬁeld does not occur and therefore case d) must be excluded here. As we are interested in a comparison of diﬀerent formulations of thermoelastic, polarizable and magnetizable bodies, we should present the constitutive theory for each case in (2.6.5). We shall discuss the details for case a) only, but the results for the two other cases will also be listed.

3.3 The Two–Dipole Models

47

Case a): ˆ iα , Pi /ρ, (µ0 Mi /ρ), Θ, Θ,i , Q) . C = C(F

(3.3.23)

Following the usual procedure, we introduce the Helmholtz free energy ψ by ψ = U − Θη , (3.3.24) and eliminate with the aid of (3.3.7)3 rext from (2.3.6). This yields d d −1 ˙ Fiα + ρEi (Pi /ρ) + ρHi (µ0 Mi /ρ) −ρψ˙ − ρη Θ˙ + tij Fαj dt dt +Ji Ei − qi,i + Θφi,i ≥ 0 .

(3.3.25)

Constitutive equations are established for the (dependent) variables tij ,

Ei ,

Hi ,

η,

φi ,

qi

and

Ji ,

which we all assume to be functions of the form (3.3.23). Furthermore, we assume the classical form of the entropy ﬂux, namely qi . (3.3.26) φi = Θ Actually this need not be done. Instead one could also postulate a general constitutive equation for φi and then prove the relation (3.3.26). For this formulation such a proof has not been given. However, in view of the equivalence of this theory with the next one (Model II) in which the result (3.3.26) was rigorously established, we may regard (3.3.26) as a proven statement. We now proceed in the way as described in Sect. 1.7. Substituting the constitutive equations for the dependent variables into the Maxwell equations (3.3.3) and into the balance laws of mass, momenta and energy (3.3.9), we arrive at the ﬁeld equations for xi , Pi , Mi , Θ and Q. Any solution to given initial data is called a thermodynamic process. Following Coleman and Noll [44], we request the entropy inequality (or any inequality derived from it) to hold for any smooth thermodynamic process. Since ρFiext and ρrext may have any arbitrarily assigned values, this implies that we may freely choose Fiα , Pi /ρ, µ0 Mi /ρ, Θ, Q, their material time derivatives and (Θ,i )· and do not violate the ﬁeld equations. Using the constitutive relation (3.3.23) for ψ , substituting the expression (3.3.26) for the entropy ﬂux φi into (3.3.25) and performing all indicated diﬀerentiations, we obtain in the inequality ˆ Pi ∂ ψˆ ∂ ψ F˙ iα + ρ Ei − tij Fαj − ρ ∂Fiα ∂(Pi /ρ) ρ µ0 Mi ∂ ψˆ +ρ H − ∂(µ0 Mi /ρ) ρ ∂ ψˆ ∂ ψˆ ˙ ∂ ψˆ ˙ qi Θ,i Θ−ρ Q + Ji Ei − ≥0, −ρ η + (Θ,i ) − ρ ∂Θ ∂Θ,i ∂Q Θ (3.3.27)

48

3 A Survey of Electromagneto-Mechanical Interaction Models

which is explicitly linear in ˙ (Θ,i )· , F˙i,α , (Pi /ρ)· , (µ0 Mi /ρ)· , Θ,

and

Q˙ .

Since all these variables may have arbitrary values, their coeﬃcients must vanish, which implies that ψˆ cannot depend on Q and Q,i , ˆ iα , Pi /ρ, µ0 Mi /ρ, Θ) , ψ = ψ(F and that Ei =

∂ ψˆ , ∂(Pi /ρ)

η=−

∂ ψˆ , ∂Θ

Hi =

∂ ψˆ , ∂(µ0 Mi /ρ)

tij = ρ

(3.3.28)

(3.3.29)

∂ ψˆ Fjα . ∂Fiα

Hence, the Helmholtz free energy cannot depend on the free charge and the temperature gradient. Of (3.3.25) there remains the residual inequality Ji Ei −

Θ,i qi ≥0, Θ

(3.3.30)

where Ji and qi are still general functions of the type (3.3.23). Since φ, η, Ei , Hi and tij must be objective scalar, vector and tensor valued ﬁelds under the full Euclidian group, ψ should have the form ˆ αβ , Pα , Mα , Θ) , ψ = ψ(C

(3.3.31)

with Cαβ , Pα and Mα as given in (2.1.10) and (2.6.7). With this choice, the constitutive equations (3.3.29) become ∂ ψˆ , ∂Θ ∂ ψˆ Ei = Fiα , ∂Pα ∂ ψˆ Hi = Fiα sgnJ , ∂Mα ∂ ψˆ tij = 2ρ Fiα Fjβ + Pi Ej + µ0 Mi Hj ∂Cαβ ! ˆ ∂ ψˆ ∂ ψˆ ∂ ψ −1 −1 =ρ 2 + Pγ Cαγ + Mγ Cαγ Fiα Fjβ . ∂Cαβ ∂Pβ ∂Mβ η =−

(3.3.32)

Conversely, it is easily shown that with this choice η, Ei , Hi and tij indeed behave as objective scalars, vectors and tensors, respectively (note that they are in correspondence with (2.6.8), (2.6.9) and (2.6.10)). Moreover, (3.3.32)4

3.3 The Two–Dipole Models

49

implies that the balance law of angular momentum (3.3.9)3 is identically satisﬁed. The above derivation makes no direct use of the Gibbs relation, as is usually done in irreversible thermodynamics. This must be so, because quite contrary to irreversible thermodynamics, the Gibbs relation is a proven statement here, valid not only in thermostatic equilibrium but also in a general thermodynamic process. Indeed, (3.3.32) together with (3.3.24) imply ∂η ∂Θ ∂η ∂Pα ∂η ∂Mα ∂η ∂Cαβ

= = = = =

1 ∂U Θ ∂Θ

∂U 1 −1 − Fαi Ei , Θ ∂Pα

∂U 1 −1 − Fαi Hi sgnJ , Θ ∂Mα

∂U 1 1 −1 −1 − (tij − Pi Ej − µ0 Mi Hj )Fαi Fβj Θ ∂Cαβ 2ρ

∂U 1 1 1 P −1 −1 − [ Tβα − (Pγ Eδ + Mγ Hδ )Cαγ Cβδ ] , Θ ∂Cαβ 2 ρ0

from which we easily obtain the so-called Gibbs relation " 1 1 P 1 −1 −1 dU − Tβα − (Pγ Eδ + Mγ Hδ )Cαγ Cβδ dCαβ dη = Θ 2 ρ0 # −1 −Eβ Cαβ dPα

−

−1 Hβ Cαβ dMα

(3.3.33)

(3.3.34)

,

P where Tαβ , Eα and Hα are deﬁned by (2.2.33) and (2.6.7), respectively. Mathematically the terms in curly brackets are called a Pfaffian form. Equation (3.3.34) delivers in a well-known manner integrability conditions for the coeﬃcient functions. These will not be derived here, because picking a particular functional representation for the Helmholtz free enˆ and evaluating the ﬁelds η, Ei , Hi and tij according to (3.3.32) ergy ψ(·) guarantees the Gibbs relation to be satisﬁed identically. We do not say that these integrability conditions must not be known in general. On the contrary, for experimentalists these relations are easier to be determined than the free energies themselves, which may then be obtained by integration. If the electromagnetic ﬁelds vanish, the above Pfaffian form reduces to

1 P 1 T dCαβ , (3.3.35) dU − dη = Θ 2ρ0 αβ

´odory [37] for thermostatic the Gibbs relation as proved correct by Carathe processes of thermoelastic materials. Irreversible thermodynamics postulates (3.3.35) to be the Gibbs relation for all those thermodynamic processes of

50

3 A Survey of Electromagneto-Mechanical Interaction Models

thermoelastic materials which deviate only slightly from thermostatic equilibrium. Here (3.3.34) holds for all thermodynamic processes in the presence of the electromagnetic ﬁelds. Needless to say that, starting with a given electromechanical interaction model, it would probably be rather diﬃcult for an irreversible thermodynamicist to guess a Gibbs relation of the form (3.3.34). Case b): We now proceed with case b), which only diﬀers from case a) in that the independent variable Pi /ρ is replaced by Ei . In this case the constitutive equations can easily be obtained from the preceding ones by the Legendre transformation 1 1 ψ˜ = ψ − Ei Pi = U − Θη − Ei Pi , ρ ρ

(3.3.36)

where ˜ iα , Ei , µ0 Mi /ρ, Θ) = ψ(C ˜ αβ , Eα , Mα , Θ) . ψ˜ = ψ(F

(3.3.37)

With (3.3.36) and (3.3.37) the constitutive equations (3.3.32) must now be replaced by ∂ ψ˜ , η =− ∂Θ ∂ ψ˜ Pi = −ρ Fiα , ∂Eα (3.3.38) ∂ ψ˜ Hi = Fiα sgnJ , ∂Mα ∂ ψ˜ tij = 2ρ Fiα Fjβ − Ei Pj + µ0 Mi Hj . ∂Cαβ With the deﬁnition 1 −1 ε˜ := U − Ei Pi = U − Eα Pβ Cαβ , ρ the Gibbs relation becomes

1 1 1 P −1 −1 dη = Tαβ + (Eγ Pδ − Mγ Hδ )Cαγ Cβδ dCαβ d˜ ε− Θ 2 ρ0 −1 −1 + Pβ Cαβ dEα − Hβ Cαβ dMα .

(3.3.39)

(3.3.40)

In a similar way we can derive constitutive equations for Case c): Here we deﬁne µ0 ¯ iα , Ei , Hi , Θ) = ψ(C ¯ αβ , Eα , Hα , Θ) . (3.3.41) ψ¯ := U − Θη − Hi Mi = ψ(F ρ The constitutive equations pertinent to this case then become

3.3 The Two–Dipole Models

∂ ψ˜ , ∂Θ ∂ ψ˜ Pi = −ρ Fiα , ∂Eα ∂ ψ˜ Mi = −ρ Fiα sgnJ , ∂Hα ∂ ψ˜ tij = 2ρ Fiα Fjβ − Ei Pj − µ0 Hi Mj . ∂Cαβ η

51

=−

(3.3.42)

Furthermore, after the introduction of 1 µ0 −1 Hi Mi = U − (Eα Pβ + Hα Mβ )Cαβ ε¯ := U − Ei Pi − , ρ ρ the Gibbs relation may be transformed into " 1 1 1 P −1 −1 Tαβ + (Eγ Pδ + Hγ Mδ )Cαγ Cβδ dCαβ dη = d¯ ε− Θ 2 ρ0 # −1 +Pβ Cαβ dEα

+

−1 Mβ Cαβ dHα

(3.3.43)

(3.3.44)

.

Before we proceed to other electromechanical interaction models, we would like to mention that all the above constitutive models are equally possible ones. Case a) is particularly fashionable among applied physicists and has for the case that the material under consideration is polarizable-only or magnetizable-only been used by, amongst others, Toupin [241] and Brown [32], respectively. The formulation of case c) is used by Pao and Hutter [171]. There are reasons for preference of case c). Indeed, since any electromagnetic problem must be solved in the entire space, including the vacuum, any theory must ultimately be expressed in terms of Ei and Hi . lt follows that in the cases a) and b) the constitutive equations for Ei and/or Hi must be invertible in the sense that Pi = fP (Fiα , Ei , Hi , Θ) , ρ

µ0 Mi = fM (Fiα , Ei , Hi , Θ) . ρ

(3.3.45)

For linear constitutive relations such inversions are trivial; they become more diﬃcult or impossible analytically when dealing with nonlinear theories. 3.3.2 The Two–Dipole Model with a Symmetric Stress Tensor (Model II) The above two-dipole model is not the only one that has been proposed. There is another one, leading to a symmetric stress tensor. This model has been investigated by Hutter [95], but it is already suggested by Fano, Chu

52

3 A Survey of Electromagneto-Mechanical Interaction Models

and Adler [73]. lt is the two-dipole counterpart of Model V, treated below. The Maxwell equations are the same as stated for model I. Electromagnetic body force, couple and energy supply, however, are taken as follows ρFie = Qe EiC + eijk Jje µ0 HkC + Qm HiC − eijk Jjm ε0 EkC , ρLeij = 0 ,

(3.3.46)

ρre = Jie Ei + Jim Hi . Here, Jie = Jie − Qe x˙ i

Jim = Jim − Qm x˙ i ,

and

(3.3.47)

and the superscripts e and m indicate that the corresponding current or charge densities are those due to electric and magnetic charges, respectively. Formally, (3.3.46)1 is simply the electric and magnetic Lorentz force. The quantities can easily be read oﬀ from the Maxwell equations (3.3.3): d PiC C Qe = Q − Pi,i , Jie = Ji + ρ − (PjC x˙ i ),j , dt ρ d µ0 MiC (3.3.48) m C m Q = −µ0 Mi,i , Ji = ρ − (µ0 MjC x˙ i ),j , dt ρ C

C

Jie = Ji + P i ,

Jim = µ0 M i .

Substitution of (3.3.48) into (3.3.46) yields ρFie = QEiC + µ0 eijk Jj HkC C +PjC Ei,j

+

C µ0 eijk x˙ j Hk,l PlC

d + ρµ0 eijk dt

C C +µ0 MjC Hi,j − ε0 eijk x˙ j Ek,l µ0 MlC

PjC ρ

d − ρε0 eijk dt

HkC µ0 MjC ρ

EkC

−(Pj Ei + µ0 Mj Hi ),j , ρLeij = 0 ,

ρre = Ji Ei + P i Ei + µ0 Mi Hi . (3.3.49) Before we proceed, let us look at the invariance properties of (3.3.49). To this end, recall that Ei , Hi , Pi = PiC , µ0 Mi = µ0 MiC , Q and Ji are objective quantities with respect to the Euclidian transformation group. Hence, the electromagnetic energy supply is an objective scalar under this group. Incidentally, it should be noted that ρre is also an objective scalar under the extended Lorentz group, provided we restrict ourselves to the semirelativistic approximation.

3.3 The Two–Dipole Models

53

Regarding the electromagnetic body force (3.3.49)1 , we see that the above expression diﬀers from (3.3.7)1 only in a term which is objective under Euclidian and under Lorentz transformations in the semi-relativistic sense. Hence, all conclusions drawn in Model I do apply also here, and we refer the reader to the pertinent discussion in Model I. In conclusion we state the governing equations of this theory consistently to within the order of exactness of the non-relativistic approximation. Hence, all terms preceded by a c−2 factor must be dropped; as a result we arrive at the following balance laws for mass, linear and angular momenta and energy: ρ˙

+ ρx˙ i,i = 0 ,

ρ¨ xi = tij,j + ρFiext + QEi + µ0 eijk Jj Hk − Pj,j Ei

+µ0 eijk P j Hk − µ0 Mj,j Hi ,

(3.3.50)

t[ij] = 0 ,

ρU˙ = tij x˙ i,j − qi,i + Ji Ei + P i Ei + µ0 Mi Hi + ρrext . Obviously the fact that ρre is by itself an objective scalar is corroboration of the assumption of symmetry of the stress tensor (compare the relevant discussion in Model I). Furthermore, a long but straightforward calculation shows that, within the non-relativistic approximation (compare these expressions with model I) II M tij

= ε0 Ei Ej + µ0 Hi Hj − 12 δij (ε0 Ek Ek + µ0 Hk Hk ) ,

II

gi = 0 ,

II

πi = −eijk EjC HkC − eijk Ej Hk − (ε0 Ek Ek + µ0 Hk Hk )x˙ i

(3.3.51)

+(ε0 Ei Ej + µ0 Hi Hj )x˙ j , II

ω = − 21 (ε0 Ek Ek + µ0 Hk Hk ) .

Hence, the jump conditions for momentum and energy become tij + ε0 Ei Ej + µ0 Hi Hj − 12 δij (ε0 Ek Ek + µ0 Hk Hk ) nj − ρx˙ i (x˙ j nj − wn ) = 0 , x˙ i tij − qj − ejkl Ek Hl + (ε0 Ej Ek + µ0 Hj Hk )x˙ k − 21 (ε0 Ek Ek + µ0 Hk Hk )x˙ j nj $ % − 12 ρx˙ i x˙ i + ρU + 12 (ε0 Ei Ei + µ0 Hi Hi ) (x˙ j nj − wn ) = 0 .

(3.3.52)

The jump conditions of electromagnetic ﬁelds and mass are the same as those for Model I (see (3.3.11)) and will not be repeated here.

54

3 A Survey of Electromagneto-Mechanical Interaction Models

We now turn to the constitutive theory. We shall not present all the details here, but list the results only. We start by eliminating ρrext from the entropy inequality (2.3.6) and the energy balance (3.3.50)4 . The result reads d Pi d µ0 Mi ˙ −ρU + ρΘη˙ + ρEi + ρHi dt ρ dt ρ (3.3.53) Θ,i qi ≥0, +(tij − Ei Pj − µ0 Hi Mj )x˙ i,j + Ji Ei − Θ where we have also used the Duhem relation for the entropy ﬂux, φi =

qi . Θ

(3.3.54)

In [95] Hutter dealt with a theory of this complexity and proved correct the form of the entropy ﬂux vector introduced here as a postulate. As before, the constitutive theory depends on the selection of the independent variables. We may again choose constitutive relations of the form (2.6.5), but we shall give them for case a) only. The equations for the other cases can be derived in the same way as was done for Model I. When the constitutive relations are of the form ˆ iα , Pi /ρ, µ0 Mi /ρ, Θ, Θ,i , Q) , C = C(F

(3.3.55)

the Helmholtz free energy ˆ αβ , Pα , Mα , Θ) , ψ := U − ηΘ = ψ(C

(3.3.56)

does not depend on Q and Θ,i and serves as potential for η, Ei , Hi and tij as follows ∂ ψˆ , ∂Θ ∂ ψˆ Ei = Fiα , ∂Pα ∂ ψˆ Hi = Fiα sgnJ , ∂Mα " # ∂ ψˆ Fiα Fjβ + E(i Pj) + µ0 H(i Mj) . tij = 2 ρ ∂Cαβ η=−

There remains Ji Ei −

Θ,i qi ≥0, Θ

as the reduced entropy inequality. Finally, the Gibbs relation takes the form

(3.3.57)

(3.3.58)

3.4 The Maxwell–Minkowski Formulation (Model III)

55

"

1 P 1 −1 −1 dU − dη = T − Cαγ Cβδ (Eγ Pδ + Eδ Pγ + Hγ Mδ + Hδ Mγ ) dCαβ Θ 2ρ0 αβ # −1 −1 −Cαβ Eβ dPα − Cαβ Hβ dMα

.

(3.3.59) For the reader’s sake of curiosity, we mention that this theory will be shown to be fully equivalent to the theory of model I. This completes the description of the two-dipole models. Both descriptions outlined above use a dipole model not only for polarization but also for magnetization. There are no others as far as we know and all the remaining models bear to a lesser or larger extent the notion of dipole structure for polarization and the structure of electric circuits for magnetization. In conclusion we wish to mention a paper by Alblas [9], who also used the Chu formulation and who derived a theory which completely agrees with our model I. The derivation is on the basis of a global energy balance law (see also Sects. 2.3 and 2.8) and for a more general problem, including spinand dissipation eﬀects. In what follows, we shall discuss theories presented by Van de Ven [249], ¨ller [160] and Hutter [92]. De Groot & Suttorp [53], Mu

3.4 The Maxwell–Minkowski Formulation (Model III) The foregoing two interaction models were constructed by postulating body force, body couple and energy supply terms due to the electromagnetic ﬁelds. The approach in this section is diﬀerent. Basically, the balance laws of mass and momenta are derived from a global energy balance law by postulating certain invariance properties. In [249] Van de Ven follows this method, ﬁrst formulated by Green and Rivlin [77], and developed for a ferromagnetic solid by Alblas [8]. By postulating that the energy balance is invariant under rigid-body motions and by making some a priori assumptions concerning the invariance properties of the quantities involved, the equations of balance of mass and momenta are derived. In this derivation all terms preceded by a factor c−2 are neglected.1 1

Clearly, the outcome of this approach essentially depends on what a priori assumptions regarding the invariance properties of the various quantities are made. Moreover, special care must be observed when neglecting terms which contain a c−2 -factor. Indeed, there is an essential diﬀerence between the approximations performed in the above mentioned references and the non-relativistic approximation in Sect. 1.6. While we apply SI-units, Gaussian units are used in [249], and it is a well-known fact that a c−2 -term in one system of units is not necessarily a c−2 -term in the other system of units as well. For instance 1 E ×H c

56

3 A Survey of Electromagneto-Mechanical Interaction Models

Before proceeding the following remark seems to be in order: It is conceivable that one might object to associate the names of Maxwell and Minkowski with the formulation presented below, and indeed neither Maxwell nor Minkowski ever presented a formulation of this kind. However, for the special case of a magnetostatic problem, the magnetoelastic stresses that will be obtained here, are identical with those used by W.F. Brown Jr. in his monograph, [32]; and Brown stated that these stresses could be derived by taking over from Maxwell a formula for the magnetic force. This is justiﬁcation for us to associate this (dynamic) theory with the name of Maxwell. Moreover, since the Maxwell equations will be used in the Minkowski formulation, we shall refer to the formulation presented in this section as the Maxwell-Minkowski formulation. Clearly, the association of names with certain theories or formulations of theories bears its well-known disadvantages and a reader not willing to accept our proposal may reject it and invent his own name for it. As an example the reader may recall that there is no unique version of the Maxwell-Minkowski stress tensor either. We shall come back to this subject at the end of this Section. As said above, the Maxwell equations will be expressed in terms of the Minkowski ﬁeld variables EiM , DiM , HiM , and BiM , which are related to the ﬁeld variables introduced in (2.2.12) and (2.2.20) by Ei = EiM + eijk x˙ j BkM ,

Di = DiM ,

Hi = HiM − eijk x˙ j DkM ,

Bi = BiM .

(3.4.1)

Accordingly, the following set of Maxwell equations is obtained: ∂Bi =0, ∂t ∂Di = Ji . − ∂t

Bi,i = 0 ,

M eijk Ek,j +

Di,i = Q,

M eijk Hk,j

(3.4.2)

Here and henceforth, since no confusion is possible, the superscript M for Di and Bi is omitted. In a purely formal way polarization per unit volume and magnetization per unit volume can be introduced via the deﬁnitions in Gaussian units becomes

1 E ×H c2 in SI-units. As a consequence of both, the a priori assumptions on invariance that are not in conformity with the results of Sect. 1.6 and the discrepancy in the neglect of terms like that demonstrated above, the equations derived in [249] are not Euclidian invariant in the sense as deﬁned in Sect. 1.6. However, it is possible to derive from an analogous energy balance as was postulated in [249] local balance equations of mass, linear and angular momentum and energy that are invariant under Euclidian transformations by simply applying in a consistent way non-relativistic invariance requirements. This then fully justiﬁes the a priori assumptions we will impose.

3.4 The Maxwell–Minkowski Formulation (Model III)

PiM := Di − ε0 EiM ,

µ0 MiM := Bi − µ0 HiM ,

57

(3.4.3)

and it is convenient to introduce also the variable Mi := MiM + eijk x˙ j PkM .

(3.4.4)

It is now a routine matter (see for instance Appendix A) to prove that under the Euclidian transformation group Ei , Di and PiM transform as objective vectors, and (3.4.5) Hi , Bi and Mi transform as objective axial vectors , respectively. In particular, and as an easy calculation also shows directly, under the rigid-body translation xi = xi − bi (t) ,

t = t

(3.4.6)

the following transformation laws must hold Di

= Di ,

EiM = EiM + eijk b˙ j Bk ,

Bi

= Bi ,

HiM = HiM − eijk b˙ j Dk ,

Q

=Q,

Ji

PiM = PiM ,

= Ji − Qb˙ i ,

(3.4.7)

MiM = MiM + eijk b˙ j PkM .

On the other hand, the Minkowski ﬁelds can also be subjected to the special Lorentz transformation (2.5.4), which when written in the semi-relativistic approximation, becomes xi = xi − Vi t ,

t = t −

1 xi Vi . c2

(3.4.8)

Then, the transformation rules are 1 eijk Vj HkM , c2 1 = Bi − 2 eijk Vj EkM , Bi c 1 = Q − 2 Ji Vi , Q c 1 M M (Pi ) = Pi − 2 eijk Vj MkM , c Di

= Di +

EiM

= EiM + eijk Vj Bk ,

= HiM − eijk Vj Dk ,

HiM

(3.4.9) Ji

= Ji − QVi ,

(MiM ) = MiM + eijk Vj PkM .

Comparison of (3.4.7) and (3.4.9), which constitute the transformation rules for special Euclidian and Lorentz transformations, shows that the primed

58

3 A Survey of Electromagneto-Mechanical Interaction Models

and stared quantities equal only in the non-relativistic approximation. This conclusion is also correct if the Minkowski-ﬁelds are subject to general Euclidian or Lorentz transformations. In view of the results of Sect. 1.6 this must be so expected. A nice application of equations (3.4.7) and (3.4.9) is obtained if the invariance properties of the Maxwell equations are investigated. To this end, they are best expressed in terms of the objective ﬁelds Ei , Di , Hi , Bi , Ji and Q and read then (see (2.2.15) and (2.2.21))

Bi,i = 0 ,

eijk Ek,j + B i = 0 ,

Di,i = Q ,

eijk Hk,j − Di = Ji .

(3.4.10)

If all the ﬁelds in these equations were independent it would follow at once from the above listed transformation properties, that the equations (3.4.10) would be invariant under Euclidian transformations. However, relations (3.4.3), which replace the Maxwell-Lorentz aether relations in the previous discussions, are not invariant under Euclidian transformations, for according to (3.4.7) and (3.4.3) we have PiM = PiM = Di − ε0 EiM = Di − ε0 EiM + ε0 eijk b˙ j Bk

1 eijk b˙ j (Hk + Mk ) , c2 + µ0 eijk b˙ j PkM = Bi − µ0 HiM + µ0 eijk b˙ j PkM

= Di − ε0 EiM +

µ0 MiM = µ0 MiM

(3.4.11)

= Bi − µ0 HiM − µ0 eijk b˙ j (Dk − Pk )

= Bi − µ0 HiM −

1 eijk b˙ j Ek . c2

Because equation (3.4.6) is a special Euclidian transformation, we conclude that relations (3.4.3) are not invariant under Euclidian transformations in general except in the non-relativistic sense. A similar calculation can also be performed with (3.4.9). The result is that (3.4.3) is a Lorentz invariant equation. We leave this proof to the reader. One further particular point in (3.4.11) must be mentioned. In reaching the non-relativistic invariance property the relation Bi = µ0 (HiM + MiM ) ,

(3.4.12)

must be used and only afterwards terms containing a c−2 -factor can be dropped. This procedure is tantamount to assuming Bi to be proportional to µ0 . Apparently this should be done consistently and hence we make the following statements that will be observed as basic rules henceforth:

3.4 The Maxwell–Minkowski Formulation (Model III)

59

(i) Bi must be considered to be proportional to µ0 . (ii) Under this provision terms containing a c−2 -factor should be dropped. When these rules are observed the relations (3.4.3) and (3.4.4) may be replaced by the very suggestive formulas Di = ε0 Ei + Pi ,

Bi = µ0 Hi + µ0 Mi ,

and

(3.4.13)

in which we have set Pi = PiM .

(3.4.14)

These preliminary remarks may suﬃce, and so we proceed to derive the balance laws of mass, linear and angular momentum and energy by starting with a global balance law of energy that is subjected to superimposed rigid-body motions (Euclidian transformations). Although our approach is diﬀerent in detail we shall in the following derivation essentially follow Van de Ven [249]. Starting equation is a global energy balance law of the form d {ρU + 12 (ε0 Ei Ei + µ0 Hi Hi ) + 12 ρx˙ i x˙ i + ρT }dν dt V

{ρr

=

ext

+

V

+ 12 (ε0 Ek Ek

ρFiext x˙ i }dν

+

{tij x˙ i − qj − ejkl EkM HlM

(3.4.15)

∂V

+ µ0 Hk Hk )x˙ j + Rj }daj .

This equation, in which ρT and Ri are still to be determined as functions of the electromagnetic ﬁeld quantities and the motion, is a generalization of the purely mechanical energy balance law. Indeed, when all electromagnetic quantities are set to zero, what results is the classical non-relativistic energy balance law. It is also easy to interpret in (3.4.15) the terms of electromagnetic origin. Firstly, 1 2 (ε0 Ei Ei + µ0 Hi Hi ) is an electromagnetic energy density, and, secondly EM × H M is the Poynting vector. As is well-known, both are not unique and we shall see that other electromagnetic models correspond to diﬀerent postulates of the electromagnetic energy density and the Poynting vector. This does not mean, however, that these models will also yield diﬀerent results for physically measurable quantities. A clue as to what should be chosen for the as yet undetermined quantities ρT and Ri is obtained, if (3.4.15) is written for a body whose mass density is vanishingly small. In this case we expect (3.4.15) to become the energy equations for the electromagnetic ﬁelds in vacuo. This energy equation is a

60

3 A Survey of Electromagneto-Mechanical Interaction Models

consequence of the Maxwell equations (3.4.10), and it can be brought into the form (3.4.15) (with vanishing tij , qi , x˙ i , U, rext , Fiext ). It is not hard to show that in this case and to within terms of order c−2 ρT = 0

and

Ri = 0 .

(3.4.16)

In other words, in a vacuum ρT and R must vanish. Next we determine ρT and R in a ponderable body, and for that purpose (3.4.15) must be written in local form. Assuming suﬃcient smoothness of the ﬁelds involved this yields (ρ˙ + ρx˙ i,i )(U + 12 x˙ i x˙ i + T ) + ρT˙ − Ri,i + (eijk EjM HkM ),i +{ρ¨ xi − ρFiext − tij,j − 12 (ε0 Ek Ek + µ0 Hk Hk ),i }x˙ i

(3.4.17)

+ρU˙ − tij x˙ i,j + qi,i − ρrext + 12 (ε0 Ek Ek + µ0 HkHk ) = 0 . This equation is now subjected to Euclidian transformations and it is postulated that it is invariant under these transformations. The details of these calculations are somewhat tedious and are presented in full detail in Appendix B. They deliver expressions for ρT and R as well as local balance laws of mass, linear and angular momentum and energy. The results for ρT and R in Appendix B are ρT = 0 ,

and

Ri = (Pj Ej + µ0 Mj Hj )x˙ i + ejkl Pk Bl x˙ j x˙ i . (3.4.18)

We note that only the second term in the expression for Ri follows from invariance requirements, whereas the ﬁrst term is chosen arbitrarily to simplify certain formulas. This seems to be a disadvantage of this method of derivation, however a change in R only results in a diﬀerent stress tensor tij (and, eventually, a diﬀerent energy ﬂux qi ) and it can be shown that this has no eﬀect in the ultimate form of the balance laws. If (3.4.18) is substituted into (3.4.15) we obtain as global energy balance d ρU + 12 (ε0 Ei Ei + µ0 Hi Hi ) + 12 ρx˙ i x˙ i dν dt V

ext ext = ρr + ρFi x˙ i dν + tij x˙ i − qj − ejkl Ek Hl V

+(Dj Ek + Bj Hk )x˙ k −

∂V 1 2 (ε0 Ek Ek

(3.4.19)

+ µ0 Hk Hk )x˙ j daj ,

from which a local balance law can be derived, which with the use of (3.4.10) and (3.4.13) may be written in the form

3.4 The Maxwell–Minkowski Formulation (Model III)

61

(ρ˙ + ρx˙ i,i )(U + 12 x˙ i x˙ i ) + ρU˙ − Ji Ei − Ei P i −µ0 Hi Mi +qi,i − ρrext − [tij + Ei Pj + µ0 Hi Mj ]x˙ i,j +[ρ¨ xi − tij,j − ρFiext − QEi − eijk Jj Bk − Pj Ej,i

(3.4.20)

−µ0 Mj Hj,i − eijk (Dj B k + Dj Bk )]x˙ i = 0 . The invariance requirements under which ρ, U, tij , (¨ xi − Fiext ), qi , ρrext , Q, Ji , Ei , Pi , Hi , Mi , Di , Bi are assumed to transform as objective quantities then yield the local balance laws of mass (3.4.21) ρ˙ + ρx˙ i,i = 0 . momentum ρ¨ xi − tij,j − ρFiext = ρFie = QEi + eijk Jj Bk

(3.4.22)

+Pj Ej,i + µ0 Mj Hj,i + eijk (Dj B k + Dj Bk ) , angular momentum t[ij] = ρLeij = P[i Ej] + µ0 M[j Hj] ,

(3.4.23)

energy ρU˙ − tij x˙ i,j + qi,i − ρrext = ρre ρre = Ji Ei + ρEi

d d (Pi /ρ) + ρµ0 Hi (Mi /ρ) . dt dt

(3.4.24)

The source terms in (3.4.22) and (3.4.24) can be transformed into the form (2.4.7) whereby, to within terms of order c−2 , III M tij

= Ei Dj + Hi Bj − 12 δij (ε0 Ek Ek + µ0 Hk Hk ) ,

III

gi = 0 ,

III

πi = −eijk Ej Hk + (Di Ej + Bi Hj )x˙ j − (ε0 Ej Ej + µ0 Hj Hj )x˙ i ,

III

ω = − 12 (ε0 Ei Ei + µ0 Hi Hi ) ,

(3.4.25)

is obtained. Substitution of (3.4.1) into (2.4.3) and of (3.4.25) into (2.4.11) yields the following set of jump conditions

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3 A Survey of Electromagneto-Mechanical Interaction Models

eijk EjM nk + Bi wn = 0 , Di ni = 0 , Bi ni = 0 , eijk HjM nk − Di wn = 0 , ρ(x˙ i ni − wn ) = 0 , tij + Ei Dj + Hi Bj − 12 δij (ε0 Ek Ek + µ0 Hk Hk ) nj − ρx˙ i (x˙ j nj − wn ) = 0 , x˙ i tij − qj − ejkl Ek Hl + (Dj Ek + Bj Hk )x˙ k − 12 (ε0 Ek Ek + µ0 Hk Hk )x˙ j nj − { 12 ρx˙ i x˙ i + ρU + 12 (ε0 Ei Ei + µ0 Hi Hi )}(x˙ j nj − wn ) = 0 .

(3.4.26)

We note that (3.4.26)6,7 can still be written in a somewhat diﬀerent form. As described in [249] it can be shown that for a material surface relations (3.4.26)6,7 are equivalent to tij nj = 12 (µ0 Mj nj )2 + (Pj nj )2 ni , (3.4.27) qj nj = x˙ i tij − 12 {(µ0 Mk nk )2 + (Pk nk )2 }x˙ j nj . If the singular surface is a boundary separating a ponderable from a body vacuum, it is easily shown that equation (3.4.27)2 reduces to qj nj = 0. In this special case, therefore, the normal component of the energy ﬂux vector must vanish. In order to make the theory complete, the balance equations must be supplemented by constitutive equations. Our presentation will be brief as it follows exactly the approach in the preceding sections. As independent variables we choose Cαβ , Eα , Mα , Θ, Θ,α

and Q ,

(3.4.28)

where Cαβ , Eα and Mα have already been introduced in (2.1.10) and (2.6.7). Deﬁning the energy functional ψ˜ by (see (3.3.36)) Pi ψ˜ := U − ηΘ − Ei , ρ

(3.4.29)

and using, as before, for the entropy ﬂux the classical relation φi =

qi , Θ

we deduce from the reduced entropy inequality that ψ˜ can neither depend on Θ,α nor Q; hence ˜ αβ , Eα , Mα , Θ) . ψ˜ = ψ(C (3.4.30)

3.4 The Maxwell–Minkowski Formulation (Model III)

63

Moreover, ∂ ψ˜ , ∂Θ ∂ ψ˜ Pi = −ρ Fiα , ∂Eα ∂ ψ˜ Hi = Fiα sgnJ , ∂Mα ∂ ψ˜ tij = 2ρ Fiα Fjβ − Ei Pj + µ0 Mi Hj . ∂Cαβ η =−

(3.4.31)

Of the reduced entropy inequality there remains the residual inequality Ji Ei −

Θ,i qi ≥0, Θ

(3.4.32)

and the Gibbs relation becomes 1 −1 dη = ] d[U − Pα Eβ Cαβ Θ 1 1 P −1 −1 − Tαβ − (Pγ Eδ + Mγ Hδ )Cαγ Cβδ dCαβ + Pα dEα − Hα dMα , 2 ρ0 (3.4.33) with deﬁnitions of Eα , Hα , Pα and Mα as given in (2.6.7). We still wish to point out that with the constitutive equation (3.4.31)4 the angular momentum equation (3.4.23) is satisﬁed identically. At this point we have set up a complete theory, consisting of balance laws, constitutive equations and jump conditions, for the interactions of electromagnetic and thermoelastic ﬁelds in solids based on the Minkowskian formulation of electrodynamics. However, there is a certain arbitrariness in the basic postulate of the theory, as manifest by the energy balance (3.4.19). As a result, the so called Maxwell stress tensor (here denoted by tM ij ) does not appear in a unique form. Therefore, we shall give here some other formulations for this global energy balance which all lead to seemingly diﬀerent theories, which, however, in the end all turn out to be equivalent. First, we wish to consider a Minkowskian formulation as given by Penfield and Haus ([177], Sect. 7.3). Taking as energy balance the expression d ρU + 12 (ε0 Ei Ei + µ0 Hi Hi ) + 12 ρx˙ i x˙ i dν dt V (3.4.34) ρrext + ρFiext x˙ i dν + tij x˙ i − qj − ejkl Ek Hl = V ∂V +(Dj Ek + Bj Hk )x˙ k − (Ek Dk + Hk Bk )x˙ j daj ,

which diﬀers from (3.4.15) or (3.4.19) only in the choice of Ri , we derive the local balance equations:

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3 A Survey of Electromagneto-Mechanical Interaction Models

ρ¨ xi − tij,j − ρFiext = ρFie

ρFie = QEi + eijk Jj Bk − Bj,i Hj − Dj,i Ej + eijk (Dj B k + Dj Bk ) = [Ei Dj + Hi Bj − δij (Ek Dk + Hk Bk )],j = tM ij,j , t[ij] = ρLeij = P[i Ej] + µ0 M[i Hj] , ∂WM + (WM x˙ i ),i − tij x˙ i,j + qi,i − ρrext = ρre ∂t ρre = Ji Ei + Ei D˙ i + Hi B˙ i ,

(3.4.35) (3.4.36) (3.4.37)

where WM := ρU + 12 (ε0 Ek Ek + µ0 Hk Hk ) .

(3.4.38)

We note that these balance laws equal those of Penfield and Haus, if the latter are taken in the non-relativistic approximation (cf. eqs. (7.39)–(7.43) of [177]). Furthermore, one can easily show that the only diﬀerence with the previous formulation lies in the stress tensor. Indeed, when we require the momentum equations in the two formulations to be the same, what results is: tij (3.4.35)−tij (3.4.22) = δij [Dk Ek +Bk Hk − 12 (ε0 Ek Ek +µ0 Hk Hk )] . (3.4.39) Once this diﬀerence is taken into account the two systems based on (3.4.19) and (3.4.34) are equivalent. Still another possible form of the energy balance is d ρU + 12 (Ei Di + Hi Bi ) + 12 ρx˙ i x˙ i dν dt V

ext ext ρr + ρFi x˙ i dν + tij x˙ i − qj − ejkl Ek Hl = V

(3.4.40)

∂V

+(Dj Ek + Bj Hk )x˙ k − 12 (Ek Dk + Hk Bk )x˙ j daj , and it leads to the following balance equations of momentum and energy ρ¨ xi − tij,j − ρFiext = ρFie ρFie = [Ei Dj + Hi Bj − 12 δij (Ek Dk + Hk Bk )],j , and

ρU˙ − tij x˙ i,j + qj,j − ρrext = ρre ρre = Ji Ei − 12 (E˙i Di − Ei D˙ i + H˙ i Bi − Hi B˙ i ) .

(3.4.41)

(3.4.42)

3.4 The Maxwell–Minkowski Formulation (Model III)

65

These equations resemble in a way a formulation in which the expressions for the electromagnetic body force and energy supply are derived from a fourdimensional formulation of the Maxwell equations, as outlined by Mφller ([158], Ch. 7) and in the report of Pao, ([172], Sect. 6). However, there is one essential diﬀerence, as in (3.4.41) and (3.4.42) the rest-frame ﬁelds Ei and Hi are used, whereas in [158] and [172] the laboratory ﬁelds Ei and Hi are employed. Since the balance laws as presented in [158] or [172] are not invariant in the non-relativistic sense, they can never be deduced from an energy balance in the way described above. This makes it very questionable whether the formulation, discussed in [158], describes the interactions between matter and ﬁeld in any meaningful way (see also [177], p. 202 and [172], pp. 121– 122). It is not too diﬃcult to show that the results corresponding to (3.4.40) and those based upon (3.4.19) are in correspondence provided that tij (3.4.40) − tij (3.4.19) = 12 δij (Ek Pk + Hk Mk ) , ρU (3.4.40) − ρU (3.4.19) = − 12 (Ei Pi + Hi Mi ) .

(3.4.43)

So far we have given three possible Minkowski formulations of electromagnetoelastic interactions. Whenever in the sequel reference is made to the Maxwell-Minkowski formulation (model III), the ﬁrst theory of this section is meant (i.e. system (3.4.21)–(3.4.24)). We conclude by stating that the two theories outlined in the preceding section could, at least for the consistent non-relativistic part, also be based upon a global energy balance. The underlying energy balance for model I is (see also Alblas [9]) d {ρU + 12 ρx˙ i x˙ i + 12 (ε0 Ek Ek + µ0 Hk Hk )}dν dt V = {ρrext + ρFiext x˙ i }dν (3.4.44) V + {tij x˙ i − qj − ejkl Ek Hl − 12 (ε0 Ek Ek + µ0 Hk Hk )x˙ j ∂V

+(ε0 Ej Ek + µ0 Hj Hk )x˙ k + (PjC Ek + µ0 MjC Hk )x˙ k }daj . In the usual way this relation leads to expressions for ρFie , ρLeij and ρre as given in (3.3.16), (3.3.7)2 and (3.3.7)3 , respectively, except for the terms preceded by a c−2 -factor in (3.3.16). In an analogous way, model II can be based upon the balance law d {ρU + 12 ρx˙ i x˙ i + 12 (ε0 Ek Ek + µ0 Hk Hk )}dν dt V (3.4.45) = {ρrext + ρFiext x˙ i }dν + {tij x˙ i − qi − ejkl Ek Hl V

∂V

− 12 (ε0 Ek Ek + µ0 Hk Hk )x˙ j + (ε0 Ej Ek + µ0 Hj Hk )x˙ k }daj .

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3 A Survey of Electromagneto-Mechanical Interaction Models

In the following sections we shall show that energy balances of this kind can also serve as bases for the theories outlined there.

3.5 The Statistical Formulation (Model IV) The proper physical approach toward a formulation of polarizable and magnetizable continua is through methods of statistical mechanics. Early attempts go back to Rosenfeldt [204]. A comprehensive treatment - in the light of relativistically covariant statistical mechanics - is given by De Groot and Suttorp [53]. This book may also serve as guideline for a historical account on the subject. In this theory, matter consists of stable groups of electrically charged particles, such as electrons, ions etc. The ﬁeld eﬀect of these particles within each stable group is represented by electric and magnetic multipoles, the statistical averages of which give rise to the deﬁnition of electric polarization Pi and magnetization Mi . With regard to the statistical derivation of the Maxwell equations, De Groot and Suttorp are entirely general and they do not introduce any nonrelativistic approximations. As a result, the macroscopic Maxwell equations are proved to be Lorentz invariant, a property hitherto assumed to hold. We introduce the statistical formulation again formally by transformation rules, viz. Bi = BiS , Ei = EiS + eijk x˙ j BkS , Di = ε0 EiS + PiS , Hi =

(3.5.1)

1 S B − eijk x˙ j ε0 EkS − MiS − eijk x˙ j PiS . µ0 i

Accordingly, the variables of the statistical formulation bear the superscript S. Substituting (3.5.1) into the Maxwell equations (2.2.15) and (2.2.21), we obtain S Bi,i =0, ∂BiS =0, ∂t S = Q − Pi,i ,

S eijk Ek,j + S ε0 Ei,i

(3.5.2)

∂E S 1 ∂PiS S S + eijk Mk,j eijk Bk,j − ε0 i = Ji + . µ0 ∂t ∂t Incidentally, Penfield and Haus [177] call this formulation the Boffi formulation and, as a comparison with (3.4.2) and (3.4.3) shows, these variables agree with those introduced in the preceding section as the Minkowski variables. The equations (3.5.2) are obtained also when the action of matter

3.5 The Statistical Formulation (Model IV)

67

upon the electromagnetic ﬁelds is derived from very crude physical models of stationary rigid bodies. Indeed they can be found in almost any physics book treating electromagnetism (see e.g. Feynman [74]). They hold, however, for a much broader class of physical processes than anticipated in many of these books and embrace all those for which the action of matter on the electromagnetic ﬁelds has been discussed above. The expressions for the electromagnetic force, couple and energy supply have been derived by De Groot and Suttorp in full generality, including relativistic eﬀects. However, they also present two approximate versions, differing in the degree of approximation, which in our terminology are called the non-relativistic and the semi-relativistic approximations. Here, we shall discuss the non-relativistic case only. In their second chapter, De Groot and Suttorp arrive at the following non-relativistic expressions for body force, body couple and energy supply (cf. [53], pp. 47, 63) d 1 S S S S Pj Bk + MjS Bj,i + ρeijk ρFie = QEiS + eijk Jj BkS + PjS Ej,i dt ρ

= QEi + eijk Jj Bk + Pj Ej,i + Mj Bj,i + eijk (P j Bk + Pj B k ) , (3.5.3)

ρLeij = P[i Ej] + M[i Bj] , d Pi ρre = Ji Ei + ρEi − Mi B˙ i , dt ρ where Bi = BiS ,

Pi = PiS

Mi = MiS + eijk x˙ j PkS .

and

(3.5.4)

Thus, the conservation of mass, the balance laws of linear and angular momenta and the balance of energy become ρ˙ + ρx˙ i,i = 0 , ρ¨ xi = tij,j + ρFiext + QEi + eijk Jj Bk + Pj Ej,i

+Mj Bj,i + eijk (P j Bk + Pj B k ) , t[ij] = P[i Ej] + M[i Bj] , ρU˙ = tij x˙ i,j − qi,i + Ji Ei + ρEi

d dt

Pi ρ

(3.5.5)

− Mi B˙ i + ρrext .

Since, as already said before, the statistical ﬁelds are identical to the Minkowski ﬁelds, we note that Q, Ji , Ei , Pi , Bi

and Mi

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3 A Survey of Electromagneto-Mechanical Interaction Models

transform under Euclidian transformations as an objective scalar, as objective vectors and as objective axial vectors, respectively. Consequently, the electromagnetic body force and body couple, as given by the above expressions, are an objective vector and a skew-symmetric tensor, respectively. Of course, this holds true also in the non-relativistic sense as deﬁned in Sect. 1.6. As far as the electromagnetic energy supply is concerned, we should note, as we already have seen in Sect. 2.2.1, that ρre cannot be an objective scalar, because the stress tensor is not symmetric in this formulation. We should rather look at the expression ρre + t[ij] x˙ i,j , which, according to (3.5.3)3 , (3.5.5)3 and (3.5.2), equals

ρre + t[ij] x˙ i,j = Ji Ei + Ei (P i +eijk Mk,j ) + (eijk Ej Mk ),i

(3.5.6)

+ (Ei Pj − Mi Bj + δij Mk Bk )x˙ (i,j) . Each term on the right-hand side of (3.5.6) is obviously an objective scalar. Hence, the balance equations are invariant under the Euclidian group. However, these laws are not invariant under Lorentz transformations, not even in the semi-relativistic sense. As the momentum equation and energy equation are invariant under Euclidian transformations it may be expected that the results given above are also derivable from a global energy balance as derived in Sect. 2.3. Indeed this is true and the underlying energy balance reads 1 d ρU + 12 ε0 Ei Ei + Bi Bi + 12 ρx˙ i x˙ i dν dt µ0 V

1 ext ext = ρr + ρFi x˙ i dν + tij x˙ i − qj − ejkl Ek Bl − Ml µ0 (3.5.7) V ∂V 1 1 Bk Bk x˙ j + ε0 Ej Ek + Bj Bk x˙ k − 21 ε0 Ek Ek + µ0 µ0 +(Pj Ek − Bj Mk + δjk Ml Bl )x˙ k daj . Using an approach analogous to that outlined in Sect. 2.3 and invoking the invariance requirement that (3.5.7) is Euclidian invariant, one can show that the above energy balance law implies the local balance equations (3.5.5). The electromagnetic body force and energy supply can also be expressed in the form (2.4.7). Indeed, straightforward calculations show that

3.5 The Statistical Formulation (Model IV)

69

1 = ε0 EiS EjS + Bi Bj + EiS Pj − MiS Bj µ0 1 − 12 δij ε0 EkS EkS + Bk Bk − 2Bk MkS + eikl Pk Bl x˙ j µ0 1 = ε0 Ei Ej + Bi Bj + Ei Pj − Mi Bj µ0 1 − 12 δij ε0 Ek Ek + Bk Bk − 2Mk Bk , µ0 IV gi = 0 , 1 S IV S πi = −eijk Ej Bk − Mk + EjS Pj x˙ i µ0 1 1 = −eijk Ej Bk − Mk − ε0 Ej Ej + Bj Bj x˙ i µ0 µ0 1 + ε0 Ei Ej + Bi Bj x˙ j + (Pi Ej − Bi Mj + δij Mk Bk ) x˙ j , µ0 1 1 1 1 IV S S ω = − 2 ε 0 Ek Ek + Bk Bk = − 2 ε0 Ek Ek + Bk Bk . µ0 µ0 (3.5.8) We note that in the above calculations (ε0 E ×B)-terms are neglected, which, according to the statement made in the preceding section between (3.4.12) and (3.4.13), is consistent with our requirements. The jump conditions for the ﬁelds, mass, momentum and energy of matter and ﬁelds become now Bi ni = 0 , ε0 EiS + Pi ni = 0 , eijk EjS nk + Bi wn = 0 , 1 S Bj − Mj nk − ε0 EiS + PiS wn = 0 , eijk µ0 ρ(x˙ i ni − wn ) = 0 , 1 Bi Bj + Ei Pj − Mi Bj tij + ε0 Ei Ej + µ0 (3.5.9) 1 1 − 2 δij ε0 Ek Ek + Bk Bk − 2Mk Bk nj µ0 − ρx˙ i (x˙ j nj − wn ) = 0 , 1 1 1 Bl − Ml − 2 ε0 Ek Ek + Bk Bk x˙ j x˙ i tij − qj − ejkl Ek µ0 µ0 1 + ε0 Ei Ej + Bi Bj x˙ i + (Ei Pj − Mi Bj + δij Mk Bk )x˙ i nj µ0

1 1 1 − ˙ i x˙ i + ρU + 2 ε0 Ei Ei + Bi Bi (x˙ j nj − wn ) = 0 . 2 ρx µ0 IV M tij

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3 A Survey of Electromagneto-Mechanical Interaction Models

It remains to develop the constitutive theory. For this purpose, we notice that, in a non-relativistic theory, the invariance group which the principle of material frame indiﬀerence relies upon, is the Galilei, or more generally, the Euclidian, group. Hence, we may choose the constitutive relations of the form derived in Sect. 1.6. In this section we restrict ourselves to the case +

C = C (Cαβ , Pα , Bα , Θ, Θ,α , Q) ,

(3.5.10)

where Pα and Bα are deﬁned in (2.6.7). The application of the entropy principle is again a routine matter and we simply state the results. With the Helmholtz free energy ψ = U − ηΘ , (3.5.11) and with (3.3.26), one ﬁnds that +

ψ = ψ (Cαβ , Pα , Bα , Θ) ,

(3.5.12)

while the entropy η, the electromotive intensity Ei , the rest-frame magnetization Mi and the stress tij are given by +

∂ψ η=− , ∂Θ +

∂ψ Ei = Fiα , ∂Pα

(3.5.13)

+

µ0 Mi = −ρ

∂ψ Fiα sgnJ , ∂Bα

+

∂ψ tij = 2ρ Fiα Fjβ + Pi Ej − Bi Mj . ∂Cαβ These results imply that the balance law of moment of momentum is satisﬁed identically. Furthermore, the reduced entropy inequality is Ji Ei −

Θ,i qi ≥0, Θ

and the Gibbs relation becomes 1 1 P 1 −1 −1 dU − dη = Tαβ + Cαγ Cβδ (Pγ Eδ − Bγ Mδ ) dCαβ Θ 2 ρ0 −1 −1 −Cαβ Eα dPβ − Cαβ Mα dBβ ,

(3.5.14)

(3.5.15)

where Eα and Mα are given in (2.6.7). This completes the non-relativistic theory of magnetizable and polarizable solids in the statistical description.

3.6 The Lorentz Formulation (Model V)

71

3.6 The Lorentz Formulation (Model V) The Lorentz description of electromagnetism is founded on his theory of electrons, originally formulated for dielectric materials only, see [244]. According to this description, the body is supposed to consist of a set of electrically interfering charged particles. These particles respond to their own ﬁelds (as well as to possible external ﬁelds) and may move rapidly, thereby producing highly ﬂuctuating (microscopic) electromagnetic ﬁelds as well. The pertinent ﬁeld equations are the Maxwell equations in a vacuum and when averaged, the Maxwell equations in the Lorentz formulation emerge (see for instance the booklet by L.Rosenfeldt [204]). In the light of De Groot and Suttorp’s statistical description this average is a crude statistical model. This might lead the reader to the conclusion that the Lorentz formulation is only approximate. This is not so. On the contrary, this theory has been put on a sound, relativistically correct, axiomatics by Truesdell and Toupin [244]. (For a non-relativistic presentation, though nevertheless relativistically ¨ller [160].) That it may be deﬁned correct formulation, see the book by Mu also from any set of Maxwell equations by simply performing a variable transformation should be corroboration of its correctness. This is exactly what we shall do here. Indeed, we introduce the Lorentz formulation by using the deﬁnitions Di = ε0 EiL + PiL ,

Ei = EiL + eijk x˙ j BkL ,

Bi = BiL ,

Hi =

1 L B − MiL − ε0 eijk x˙ j EkL . µ0 i

(3.6.1)

Upon substitution of (3.6.1) into (2.2.15) and (2.2.21), the Maxwell equations in the Lorentz formulation are obtained as follows: L Bi,i =0,

∂BiL =0, ∂t L = Q − Pi,i ,

L eijk Ek,j + L ε0 Ei,i

(3.6.2)

∂E L 1 ∂PiL L L + eijk (eklm PlL x˙ m ),j + eijk Mk,j eijk Bk,j − ε0 i = Ji + . µ0 ∂t ∂t These equations, introduced here formally, are often applied in the modern ¨ller [127], Benach literature, as for instance by Toupin [241], Liu and Mu ¨ller [25] and Hutter [92, 95, 96]. The equations are also mentioned and Mu by Pao [172], but he states “that for magnetizable materials their validity is questionable”. Comparing (3.6.2) with the Maxwell equations of the statistical model (3.5.2), we see that

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3 A Survey of Electromagneto-Mechanical Interaction Models

EiL = EiS ,

BiL = BiS = Bi ,

PiL = PiS = Pi ,

MiL = MiS + eijk x˙ j PkL = Mi .

(3.6.3)

The Maxwell equations in the Lorentz description can also be deduced from the assumption that every particle is equipped with a number of electric dipoles and with an electric circuit. If this dipole-circuit model is treated nonrelativistically, again the Maxwell equations in the Lorentz description are obtained. Of course, such a derivation bears inherently the notion of approximation. What is approximated thereby is not the theory as such, however, but the use of the model. That the theory is indeed correct from a relativistic point of view cannot be seen from such a derivation, and this must be regarded as a major disadvantage of the model. This might also be the reason why Penfield and Haus use a model in which each particle is equipped with a number of dipoles and an electric circuit both of which are treated relativistically. What they obtain is diﬀerent from the Lorentz `rean model. The description. They call these Maxwell equations the Ampe `rean and the Lorentzean variables is a small diﬀerence between the Ampe diﬀerence in the polarization vector of the amount c−2 (x˙ × M ) . In a non-relativistic theory this term is negligible, and it emerges from the fact that in the Lorentz description PiL = PiM , `rean description while in the Ampe PiA = PiC , `rean and the Lorentz formulation are relaNevertheless, both the Ampe tivistically correct. This should make it clear that the approximations must be sought in the model rather than in the basic theory. With this digression on approximations we proceed and mention that in the Lorentz formulation L − Pi,i is the charge density due to polarization, L eijk Mk,j

the current due to magnetization and ∂PiL + eijk (eklm PlL x˙ m ),j ∂t the polarization current, all as shown on the right-hand side of (3.6.2). This suggests to postulate as electromagnetic body force, body couple and energy supply the expressions

3.6 The Lorentz Formulation (Model V)

73

ρFie = Qtot EiL + eijk Jjtot Bk = Qtot Ei + eijk Jjtot Bk , ρLeij = 0 ,

(3.6.4)

ρre = Jitot Ei Jitot EiL − ρFie x˙ i , where Qtot and Jitot are the total charge and total current densities as given on the right-hand side of (3.6.2)3,4 . (Compare this postulate with that for model II, (3.3.46)). The postulates (3.6.4) have, in this generality, ﬁrst been ¨ller introduced by Toupin [241]. They were then applied by Liu and Mu ¨ller [25], and Hutter [92] in theories of polarizable [127], Benach and Mu and magnetizable ﬂuids, ﬂuid mixtures and solids, respectively. Substituting Qtot and Jitot , as obtained from (3.6.2)3,4 , into (3.6.4) yields

ρFie = (Q − Pj,j )Ei + eijk (Jj + P j +ejmn Mn,m )Bk , ρLeij = 0 ,

(3.6.5)

ρre = Ji Ei + Ei (P i +eijk Mk,j ) , where the convective derivative, denoted by a superimposed star, is deﬁned in (2.2.17). With (3.6.5) the balance laws of mass, momentum and energy assume the form ρ˙ + ρx˙ i,i = 0 ,

ρ¨ xi = tij,j + ρFiext + (Q − Pj,j )Ei + eijk (Jj + P j +ejmn Mn,m )Bk ,

(3.6.6)

t[ij] = 0 ,

ρU˙ = tij x˙ i,j − qi,i + Ji Ei + Ei (P i +eijk Mk,j ) + ρrext . The expressions (3.6.5) immediately show that ρre and ρFie are an objective scalar and objective vector under the Euclidian transformation group. For this to be true, we have used that Q, Ji , Ei , Bi , Pi and Mi are objective quantities, as we have already seen in the preceding sections. A proof of this can also be found in Truesdell and Toupin [244], Chap. 7. Hence, the momentum and energy equations enjoy the classical (non-relativistic) invariance requirements. The energy equation is, however, only invariant under Euclidian transformations, because we have also taken the stress tensor to be symmetric. This may serve as a justiﬁcation for the rather unmotivated choice of ρLeij = 0. Just as in the preceding sections, this formulation can also be derived from a global energy balance which in this case reads

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3 A Survey of Electromagneto-Mechanical Interaction Models

d 1 1 1 Bk Bk ρU + 2 ρx˙ i x˙ i + 2 ε0 Ek Ek + dν dt µ0 V ρrext + ρFiext x˙ i dν = V

Bl + + tij x˙ i − qj − ejkl Ek µ0

1 2

(3.6.7)

ε0 Ek Ek +

1 Bk Bk x˙ j µ0

daj .

∂V

Furthermore, it may be shown that the expressions (3.6.5) can be written in the form (2.4.7) (with the non-relativistic approximations), yielding 1 1 V M tij = ε0 Ei Ej + Bi Bj − 12 δij ε0 Ek Ek + Bk Bk , µ0 µ0 V gi = 0 , Bk πi = −eijk EjL = −eijk Ej (Hk + Mk ) µ0 1 1 Bk Bk x˙ i + ε0 Ei Ej + Bi Bj x˙ j , − ε0 Ek Ek + µ0 µ0 1 1 V ω =− Bk Bk . ε0 Ek Ek + 2 µ0 V

(3.6.8)

With these relations, the jump conditions for momentum and energy of matter and ﬁelds can be derived. We list them below, together with the jump conditions for the electromagnetic ﬁelds and the density Bi ni = 0 , ε0 EiL + Pi ni = 0 , eijk EjL nk + Bi wn = 0 , 1 Bj − Mj + ejlm x˙ l Pm nk − ε0 EiL + Pi wn = 0 , eijk µ0 ρ(x˙ i ni − wn ) = 0 , 1 1 1 Bi Bj − 2 δij ε0 Ek Ek + Bk Bk nj tij + ε0 Ei Ej + µ0 µ0 − ρx˙ i (x˙ j nj − wn ) = 0 , Bl 1 − 12 ε0 Ek Ek + Bk Bk x˙ j tij x˙ i − qj − ejkl Ek µ0 µ0 1 + ε0 Ei Ej + Bi Bj x˙ i nj µ0

1 1 1 − ˙ i x˙ i + ρU + 2 ε0 Ek Ek + Bk Bk (x˙ j nj − wn ) = 0 . 2 ρx µ0 (3.6.9)

3.6 The Lorentz Formulation (Model V)

75

We wish to note that the jump conditions (3.6.9)6,7 and all corresponding ones of the preceding sections, could have been derived from the respective global energy balance laws. To illustrate this, note that from the global energy balance (3.6.7) a jump condition can be derived as demonstrated in Sect. 1.5. This jump condition (given by (3.6.9)7 ) is then subjected to a rigid-body translation of the form xi ⇒ xi − bi (t) , under which (x˙ j nj −wn ) remains invariant. Since under such transformations − eijk EjL Bk /µ0 changes into (recall that although BiL is invariant, BiL /µ0 is not (see Sect. 1.6))

Bk 1 1 − ε0 Ei Ej + Bi Bj − δij ε0 Ek Ek + Bk Bk bi , − eijk EjL µ0 µ0 µ0 (3.6.9)7 then immediately leads to (3.6.9)6 . We now turn our attention to the constitutive theory and for that purpose it is advantageous to transform the energy balance equation (3.6.6)4 into the form d Pi ˙ ρU = tij x˙ i,j − qi,i + Ji Ei − (eijk Ej Mk ),i + ρEi dt ρ (3.6.10) ext ˙ −Mi Bi − (Ei Pj − Mi Bj + Mk Bk δij )x˙ i,j + ρr . Then, elimination of ρrext from (2.3.6) and introduction of the Helmholtz free energy by ψ = U − ηΘ , reveals the inequality d Pi ˙ −ρψ − ρΘη˙ + ρEi − Mi B˙ i + {tij − Ei Pj + Mi Bj dt ρ −Mk Bk δij }x˙ i,j + Ji Ei + [Θφi,i − qi,i − (eijk Ej Mk ),i ] ≥ 0 .

(3.6.11)

Before we proceed, a comment seems to be in order regarding the last, bracketed, term of this inequality. Evidently, there is no unique deﬁnition of energy ﬂux and entropy ﬂux in terms of thermodynamic and electromagnetic variables. To see this, let us introduce a new energy ﬂux vector qiS by qiS := qi + eijk Ej Mk .

(3.6.12)

As will be shown in the next chapter, qiS agrees with the energy ﬂux vector of the statistical model IV. With the choice

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3 A Survey of Electromagneto-Mechanical Interaction Models

φi = qiS /Θ ,

(3.6.13)

then becomes [−Θ,i qiS /Θ], one insisted to call qiS the

the bracketed term in (3.6.11) and this is what heat ﬂux vector. one would expect classically, if However, this is a purely formal deﬁnition, and no argument whatsoever justiﬁes such an identiﬁcation. Moreover, if we substitute (3.6.12) into the classical relation for the entropy ﬂux (3.3.26), we ﬁnd 1 qi + eijk Ej Mk , (3.6.14) Θ Θ and, if we insist to call qi the heat ﬂux, this equation teaches us that in this case the entropy ﬂux does not obey the classical relation, namely heat ﬂux divided by absolute temperature. This is the reason why we prefer to call qi the energy ﬂux vector. Moreover, this shows that it is imperative to refrain from setting the entropy ﬂux equal to the heat ﬂux divided by absolute temperature from the outset, because it is not evident what should be understood under heat ﬂux. Instead, one must prescribe φi as a general constitutive variable, whose form should be determined in due course with the exploitation of the entropy inequality. For one particular theory of the complexity treated here, the relation φi =

qi + eijk Ej Mk qiS = , (3.6.15) Θ Θ was proved by Hutter [92], and the same relation was also found earlier by ¨ller [127] for the case of a simple ﬂuid. For the purpose of this Liu and Mu monograph we shall regard (3.6.15) as a postulate. As was done for the other models, we now establish constitutive relations, which in view of (3.6.11) are assumed in the form φi =

+

C =C (Fiα , Pi /ρ, Bi , Θ, Θ,i , Q) ,

(3.6.16)

where C denotes the set {U, ψ, η, Ei , µ0 Mi /ρ, qi , Ji , tij }. Moreover, we require the constitutive relations to be invariant under the Euclidian group of transformations. In view of the transformation rules listed before this is already ascertained. In the usual way we can derive the following results: +

ψ =ψ (Cαβ , Pα , Bα , Θ) , and

(3.6.17)

+

∂ψ η=− , ∂Θ +

∂ψ Ei = Fiα , ∂Pα +

∂ψ µ0 Mi =− Fiα sgnJ , ρ ∂Bα

(3.6.18)

3.7 Thermostatic Equilibrium – Constitutive Equations

77

+

∂ψ tij = 2ρ Fiα Fjβ + (Pi Ej + Pj Ei ) − (Mi Bj + Mj Bi ) + δij Mk Bk . ∂Cαβ The reduced entropy balance then reads Ji Ei −

Θ,i (qi + eijk Ej Mk ) ≥ 0 , Θ

and the Gibbs relation becomes

1 1 1 P −1 −1 dη = T − Cαγ Cβδ (Pγ Eδ + Pδ Eγ ) dU − Θ 2 ρ0 αβ −1 −1 −1 −Cαγ Cβδ (Mγ Bδ + Mδ Bγ ) − Cαβ Mγ Mγ dCαβ −1 −1 −Cαβ Eβ dPα + Cαβ Mβ dBα .

(3.6.19)

(3.6.20)

This completes the presentation of the Lorentz formulation.

3.7 Thermostatic Equilibrium – Constitutive Equations for Electric Current and Energy Flux In the preceding sections we determined constitutive equations for entropy, two electromagnetic ﬁeld vectors and the stress tensor. All of the above mentioned variables are derivable from a thermodynamic potential (the latter being the Helmholtz free energy or one of its Legendre transformations). Given such a thermodynamic potential as a function of its independent variables the constitutive relations for the remaining ﬁelds are derivable. However, it is not possible to construct in the same way constitutive relations for the conductive current density and for the energy ﬂux. Constitutive relations for these quantities are merely restricted by the residual entropy inequality, which in all but the Lorentz formulation may be written as γ := Ji Ei −

Θ,i qi ≥0. Θ

(3.7.1)

In the Lorentz formulation qi must be replaced by qiS := qi + eijk Ej Mk .

(3.7.2)

To ﬁnd the restrictions imposed by the residual entropy inequality we look at thermostatic processes, which will be deﬁned as processes with uniform and time-independent temperature and vanishing conductive current. The results depend on whether we are dealing with an electrical insulator or conductor. Hence, we discuss these cases separately.

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3 A Survey of Electromagneto-Mechanical Interaction Models

(a) In an electrical insulator the conductive current Ji vanishes identically. Hence, (3.7.1) becomes Θ,i qi ≥0, (3.7.3) γ=− Θ and in equilibrium & & γ = γ& = 0 . (3.7.4) E & & We shall characterize thermostatic equilibrium by the index & . It follows E that γ, which is a function of all the independent variables upon which qi depends, assumes its minimum when Θ,i = 0. Necessary conditions for this to be the case are ∂γ && and & =0, ∂Θ,i E (3.7.5) ∂ 2 γ && is positive-semi deﬁnite , & ∂Θ,i ∂Θ,j E or

& & qi & = 0 , E

∂q(i && & ∂Θ,j) E

and (3.7.6) is negative-semi deﬁnite.

Hence, for an electrical insulator in thermostatic equilibrium the energy ﬂux vector must necessarily vanish. To see what consequences these restrictions impose, let us consider the case for which the constitutive relations are given in the form ˇ iα , Ei , Bi , Θ, Θ,i , Q) . C = C(F

(3.7.7)

A constitutive relation for the energy ﬂux vector that is objective under Euclidian transformations and depends on the above variables is of the form (3.7.8) qi = Fiα qˇα (Cβγ , Eγ , Bγ , Θ, Θ,β , Q) , where Eγ , Bγ and Cβγ have been deﬁned in (2.6.7) and where Θ,α := Fiα Θ,i .

(3.7.9)

A necessary and suﬃcient condition for the energy ﬂux vector to vanish in thermostatic equilibrium is that qi is of the form qi = −Fiα kˇαβ (Cγδ , Eγ , Bγ , Θ, Θ,γ , Q)Θ,β .

(3.7.10)

3.7 Thermostatic Equilibrium – Constitutive Equations

79

This representation is suitable for solids to which we shall restrict ourselves henceforth. In an electrical insulator, (3.7.10) is also the most general constitutive relation for the energy ﬂux vector which vanishes in thermostatic equilibrium. For this form of the constitutive relation, the condition (3.7.6)2 requires that the matrix & & Aij := F(iα Fj)β kˇαβ & is positive-semi deﬁnite. (3.7.11) E

In the theory of solids one often restricts oneself to small deformations and small deviations from thermostatic equilibrium. In such special cases, in the constitutive equations all terms that are quadratic or of higher order in the above mentioned small quantities are neglected. Since the constitutive relation for qi is already explicitly linear in Θ,i this requires that kˇαβ is independent of Cαβ and Θ,α and that Fiα may be replaced by δiα . For such a case (3.7.11) reduced to & & is negative-semi deﬁnite. (3.7.12) Aij := kˇ(ij) & E

Nothing can be concluded from the equilibrium condition about the skew& & ˇ symmetric part of kij & . As a consequence of the Onsager relations, one E usually requires that & & (3.7.13) kˇ[ij] & = 0 . E

Finally, in this linear approximation it is not diﬃcult to show that qi = −kˇij (Ek , Bk , Θ, Q)Θ,j .

(3.7.14)

Here, as a consequence of the Onsager relations, we require kˇ[ij] = 0 ,

(3.7.15)

kˇij Θ,i Θ,j ≥ 0 .

(3.7.16)

and, furthermore, In this drastically simpliﬁed version the constitutive relation (3.7.14) for the energy ﬂux vector is known as the Fourier law of heat conduction. (b) In an electrical conductor one may write qi = qˇi (Fjα , Ej , Bj , Θ, Θ,j , Q) , Ji = Jˇi (Fjα , Ej , Bj , Θ, Θ,j , Q) ,

(3.7.17)

and then γ is a function of Fiα , Ei , Bi , Θ, Θ,i

and Q .

Since the electrical insulator as a special case of an electrical conductor with zero conductivity was already considered on the previous pages,

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3 A Survey of Electromagneto-Mechanical Interaction Models

we exclude it here. For the general case, the second of the constitutive relations (3.7.17) may be inverted in the sense that Ei = Eˇ i (Fjα , Jj , Bj , Θ, Θ,j , Q) .

(3.7.18)

If this relation is substituted into (3.7.17)1 , we may regard γ to be a function of Fiα , Ji , Bi , Θ, Θ,i and Q , so that (3.7.1) becomes γ = Ji Eˇ i − Since then

Θ,i ˇ q = γˇ (Fiα , Ji , Bi , Θ, Θ,i , Q) ≥ 0 . Θ i

& & & & & & & & & & =0, γ & = γˇ Fiα & , 0, Bi & , Θ& , 0, Q& E

E

E

E

E

(3.7.19)

(3.7.20)

we must necessarily have ∂ γˇ && & =0, ∂Ji E and



∂ 2 γˇ  ∂J ∂J i j       ∂ 2 γˇ ∂Ji ∂Θ,j

 ∂ 2 γˇ ∂Ji ∂Θ,j       2ˇ ∂ γ  & ∂Θ,i ∂Θ,j &&

∂ γˇ && & =0, ∂Θ,i E

(3.7.21)

is positive-semi deﬁnite .

(3.7.22)

& & and qˇi & = 0 .

(3.7.23)

E

Of necessity then & & Eˇ i & = 0 , E

E

Therefore, the electromotive intensity and the heat ﬂux vector vanish in thermostatic equilibrium. Hence, we could have deﬁned thermostatic equilibrium also as a time-independent process for which Ei and Θ,i vanish. Then, the above relations would read ∂ˇ γ && & 0, ∂Ei E and

∂ˇ γ && & =0, ∂Θ,i E

(3.7.24)

3.7 Thermostatic Equilibrium – Constitutive Equations



∂ 2 γˇ  ∂E ∂E  i j      ∂ 2 γˇ ∂Ei ∂Θ,j

81



∂ 2 γˇ ∂Ei ∂Θ,j       ∂ 2 γˇ  & ∂Θ,i ∂Θ,j &&

is positive-semi deﬁnite .

E

The relations (3.7.24), (3.7.25) now yield & & & & Jˇi & = 0 , qˇi & = 0 , E

as well as  ∂ Jˇ(i  ∂Ej)       ∂ Jˇi 1 ∂ qˇj − ∂Θ,j Θ ∂Ei

(3.7.25)

E

 ∂ Jˇi 1 ∂ qˇj − ∂Θ,j Θ ∂Ei        1 ∂ qˇ(i & − & Θ ∂Θ,j) &

(3.7.26)

is positive-semi deﬁnite .

E

(3.7.27) The most general form of the constitutive equations (3.7.17) for a solid satisfying the principle of material frame indiﬀerence is ˇα (Cβγ , Eβ , Bβ , Θ, Θ,β , Q) , qi = Fiα q Ji = Fiα ˇJα (Cβγ , Eβ , Bβ , Θ, Θ,β , Q) .

(3.7.28)

When they are written in the form (q) qi = Fiα {−kˇαβ Θ,β + βˇαβ Eβ } , (J ) Ji = Fiα {βˇαβ Θ,β + σ ˇαβ Eβ } ,

(3.7.29)

they automatically satisfy the equilibrium conditions (3.7.26). Moreover, in order to satisfy (3.7.27), the two relations (3.7.29) imply that   ˇ(J ) − 1 βˇ(q) )Fiα Fjβ σ ˇ F F ( β αβ (iα j)β αβ   Θ βα     (3.7.30) A :=       1ˇ 1 (q) (J ) (βˇαβ − βˇβα )Fiα Fjβ kαβ F(iα Fj)β Θ Θ must be positive-semi deﬁnite. Necessary conditions (that are not suﬃcient however) for this to be satisﬁed are

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3 A Survey of Electromagneto-Mechanical Interaction Models

σ ˇαβ F(iα Fj)β

and

1ˇ kαβ F(iα Fj)β Θ

are positive-semi deﬁnite .

(3.7.31) More interesting than the general case is again the linearized version of the theory. Assuming small deformations and small deviations from thermostatic equilibrium, it may be justiﬁed to postulate the constitutive relations for qi and Ji , (3.7.29), to be linear in Θ,i and Ei and independent (J ) (q) of the deformation. In this case, the coeﬃcient functions kˇαβ , βˇαβ , βˇαβ , and σ ˇαβ are independent of Cαβ , Eα and Θ,α , and Fiα may be replaced by δiα . Under these simpliﬁed restrictions   1 ˇ(q) (J ) ˇ σ ˇ(ij) βij − βji   Θ     (3.7.32) A≈ ,     1ˇ 1 (q) (J ) βˇij − βˇji k(ij) Θ Θ must be positive-semi deﬁnite. Of necessity then, σ ˇ(ij) is positive deﬁnite (because σ ˇ ≡ 0 is excluded) and kˇ(ij) is positive-semi deﬁnite. The Onsager relations require here that σ ˇ[ij] = 0 ,

kˇ[ij] = 0 ,

1 (q) 1 (J ) βˇij = βji =: βji . Θ Θ

(3.7.33)

Of course, σ ˇij , kˇij and βˇij are still functions of Bk , Θ and Q. When linearized and with the use of (3.7.33), the constitutive equations (3.7.29) become qi = −kˇij (Bk , Θ, Q)Θ,j + βˇij (Bk , Θ, Q)Ej , Θ,j Ji = βˇji (Bk , Θ, Q) +σ ˇij (Bk , Θ, Q)Ej . Θ

(3.7.34)

These equations are known as Fourier’s law of heat conduction and Ohm’s law of electrical conduction, and in this form they appear in usual treatises on crystallography and solid state physics (e.g. [165, 29]). The coeﬃcients ˇαβ are termed the thermal and electrical conductivity, respectively. kˇαβ and σ The coeﬃcients βˇαβ are responsible for thermoelectric eﬀects (e.g. Seebeck eﬀect, Peltier heat, Thomson heat, cf. [165], Chap. 12 or [263], Chap. 23), and in magnetizable materials they also cause galvanomagnetic or thermomagnetic eﬀects (e.g. Ettinghausen or Nernst eﬀect). These eﬀects are investigated in more detail than is possible within the scope of this monograph, in a series of papers by Pipkin and Rivlin, [180] and [181], and Borghesani and Morro, [27] and [28]. Pipkin and Rivlin derive general, nonlinear conduction laws for isotropic materials considering separately the

3.8 Discussion

83

eﬀects of deformation on the electric current and the inﬂuences of the electric ﬁeld, the magnetic induction and the temperature gradient on the constitutive equations for the electric current, the heat ﬂux and the magnetic ﬁeld strength, whereas Borghesani and Morro derive the fourth-order versions of Fourier’s law and Ohm’s law. Finally we remark that for most purposes of this monograph the consequences implied by the Onsager relations will be of no relevance. In particular, the results of Chap. 3 are independent of such relations. When using the Onsager relations we shall in the following therefore explicitly state it. In summary, we have given an exposure of ﬁve diﬀerent descriptions of deformable polarizable and magnetizable continua. Each model describes in its own way the interactions between the electromagnetic ﬁelds and the thermoelastic body. The ﬁeld equations consist of the following set of equations: (i) The Maxwell equations, which must be counted as seven independent equations (see the remark in Sect. 1.3.2). (ii) Five balance laws of mass, momentum and energy. (iii) Sixteen constitutive relations for entropy, two electromagnetic ﬁeld vectors and the stress tensor (the latter given in a form such that the balance law of moment of momentum is satisﬁed identically). (iv) Six constitutive relations for the electric current and the energy ﬂux vector. These are 34 equations for the unknowns: four electromagnetic ﬁeld vectors (12), electric current and electric charge density (4), mass density (1), motion χi (3), stress (9), temperature (1), energy ﬂux (3) and entropy (1). These ﬁeld equations are supplemented by jump (or boundary) conditions for the electromagnetic ﬁeld variables, mass, momentum and energy of matter and ﬁeld.

3.8 Discussion In the preceding sections ﬁve diﬀerent descriptions of deformable polarizable and magnetizable continua were treated. The interaction of the electromagnetic ﬁelds with matter was achieved by introducing two additional electromagnetic ﬁeld vectors to the two basic ﬁelds occurring in vacuo. In the ﬁve formulations the choice of these ﬁeld vectors was not unique, however. We did not emphasize the models which lie behind these descriptions, although each of the formulations can be based upon well-deﬁned models. The Chu variables of electrodynamics for instance can be founded on a two-dipole model, and as long as one restricts oneself to the derivation of the Maxwell equations, such a model leads to unique answers. The model may bear its disadvantages insofar as it is handled non-relativistically, while the resulting equations are postulated to be relativistically correct equations. This does not change the basic fact that the resulting Maxwell equations are unique.

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3 A Survey of Electromagneto-Mechanical Interaction Models

When electromagnetic body force and energy supply are derived, however, the dipole model is no longer unique and two theories of magnetizable and polarizable bodies can be developed. We have seen this when presenting models I and II. One can also develop a theory on the basis that polarization is modelled by a dipole while magnetization is treated as an electric circuit. Dependent on the degree of complexity of derivation, diﬀerent Maxwell equations emerge (statistical and Lorentz formulation), and also the expressions for the electromagnetic body force, body couple and energy supply are diﬀerent in these derivations. The reader could therefore be misled by these derivations as he might conclude that some of the theories are superior to others, because the derivation of the equations resembles a more profound approach. This is not so, and our point of view is diﬀerent, as we regard all formulations as equally sound as long as none has been proved to be superior to any other one, e.g. via transformation properties under Euclidian or Lorentz transformations. Of course, this requires that we de-emphasize the models behind the equations. This is why we do not even share the viewpoint of many physicists, who would reject the two-dipole description on the basis that magnetic monopoles have never been observed experimentally and that therefore magnetization must be electric circuits. Corroboration to all the above statements will be found in the next chapter of this monograph. For instance, the statistical model and the Lorentz model (models IV and V) can be derived by methods of statistical physics, but dependent on which author’s book one opens, either one appears to be an approximation of the other. The reason that in the most rigorous treatment the Lorentz formulation is not obtained is a matter of deﬁnition of the macroscopic electric and magnetic dipole moments in terms of statistical averages of microscopic quantities. Nonetheless, Pao concludes in his reviewing article [172] that for magnetizable materials the validity of the Lorentz model is questionable (page 24 of [172]). Yet, we shall prove that the non-relativistic versions of polarizable or magnetizable continua in the models IV and V are equivalent. A similar situation also exists for the expression of the body force as suggested by Fano, Chu and Adler that agrees with our model II. Again according to Pao (page 82 of [172]), “Penfield and Haus [177] have considered these force expressions incomplete”. We shall prove that models I and II are equivalent. Needless to say that Penfield and Haus [177] and Pao and Hutter [171] have advocated for model I. The above statements should make it clear then that the derivation of the models is not important and that the formulations should be uniﬁed by demonstrating their equivalence, rather than emphasizing their diﬀerences. For this reason we have presented the ﬁve models per se and have studied their invariance properties quite extensively. This led to the approach of model III, in which electromagnetic body force, body couple and energy supply were derived from a general energy expression that is subjected to speciﬁc invariance requirements under the general Euclidian transformation group. In order to

3.8 Discussion

85

illustrate the above mentioned uniﬁcation and because some authors seem to have objections against the approach of Sect. 2.3 (as they do not believe that this approach will lead to unique results) we have also given the energy laws which could serve as basis for the other models. For model I this was already done by Alblas [9]. This, together with the results of Chap. 3, in which all models are shown to be equivalent, demonstrates that there is a large amount of freedom in the choice of the speciﬁc terms occurring in such an energy law, all leading to equivalent results. We found further that in formulations in which the stress tensor was symmetric the electromagnetic body force and energy supply are objective quantities with respect to the Euclidian transformation group. In formulations with non-symmetric stress, body force and body couple turned out to be objective, while the energy supply term ﬁrst had to be corrected by the power of working of the skew-symmetric part of the stress. As a result, the energy equation remained invariant. These transformation properties make it possible that the various formulations of electromagnetic interactions with thermoelastic bodies have a chance to be equivalent. Indeed, full equivalence cannot be achieved when corresponding quantities in diﬀerent formulations do not transform alike under the Euclidian transformation group. Similarly, if approximations in the non-relativistic sense are performed, equivalence can only be attempted to be proved within such approximations. This discussion would not be complete if we would not point out that other descriptions for ﬁeld matter interactions often agree with our models I-V. We ﬁnd it important to discuss these descriptions also. In order to narrow the number of formulations down to a reasonable size, we shall not discuss any quasi-static formulation, such as those by Toupin [240], Alblas [8], Tiersten [233, 236], Pao and Yeh [170] and others (see e.g. Penfield and Haus [177]), with the exception of the monograph of Brown [32], however. The reason for this exception is that Brown already was aware of the non-uniqueness of the magnetoelastic stresses, and he stated explicitly that these stresses are only then completely determined once the total system of momentum and moment of momentum equations, boundary conditions and constitutive equations are given. For a magnetostatic theory, Brown introduced four diﬀerent stress tensors, namely: (see Sect. 5.6 of [32]) (i) (ii) (iii) (iv)

The The The The

pole-model t¯ij (eqs. (6.2.9)–(6.2.12)). `rean current model t¯ij (eqs. (6.2.9 )(6.2.12 )). Ampe Maxwell model I tij (eqs. (6.2.17)–(6.2.23)). Maxwell model II tij (eqs. (6.2.17 )-(6.2.23 )).

Comparing these stress models with the ones introduced in this chapter, we see that when the latter are simpliﬁed to the magnetostatic case, the following relations between our stresses and those of Brown hold:

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3 A Survey of Electromagneto-Mechanical Interaction Models I

tij =

III

tij = tij ,

II

tij = t¯ij , (3.8.1)

IV

tij =

tij

,

V

tij =

t¯ij

.

In comparing other authors’ work, any complications due to spin interactions will be left aside. This does not mean that spin interaction theories will not be compared here, but any contribution due to magnetic spin will be set to zero without further mentioning. Furthermore, fully relativistic theories will only be mentioned in connection with non-relativistic approximations. First when the material is polarizable-only, model V is in full agreement with a description of a dynamical theory of elastic dielectrics as presented by Toupin [241], and indeed it must be so as our force and energy supply expressions agree with those of Toupin. On the other hand, Dixon and Eringen attempted to derive a theory of ﬁeld-matter interaction using a statistical description (which we believe to be incomplete) [59]. They include in their derivation also macroscopic electric quadrupoles, but when these are neglected their Maxwell equations and body force and energy supply expressions are in agreement with model V. However, the jump condition of energy derived by Dixon and Eringen (their equation (5.16)) is not correct. Indeed, their global energy balance contains a volume source which becomes indeﬁnitely large when a surface of discontinuity is approached. Hence, apart from these shortcomings in the derivation, their results are non-relativistically correct. Moreover, Eringen in collaboration with Grot [78], and Grot [79] present relativistic formulations of solids and ﬂuids in the electromagnetic ﬁelds, in which they postulate an energy-momentum tensor of ﬁeld-matter interaction which reduces to the force and energy expressions of model V when terms of O(V 2 /c2 ) are neglected. Of a quite diﬀerent nature is Tiersten’s approach to describe the interaction of the electromagnetic ﬁelds with deformable continua. He uses a mixture concept and describes the ﬁeld-matter interactions by coupling a so called lattice-continuum with charge-, electronic-, ionic- and spin-(sub)continua, each bearing the notion of a particular physical eﬀect. All work of Tiersten, [236], is based on such a mixture concept. In [237] Tiersten and Tsai describe the interaction of the electromagnetic ﬁelds with heat conducting deformable electric insulators. In a simpliﬁed version of this theory, where only a charge continuum is interacting with the lattice continuum, their body force, body couple and energy supply (see equations (3.48), (8.4) and (8.7) in [237]) agree with model IV. An explicit proof for this equivalence can also be found in Pao’s review article, [172]. The same comments hold for the paper of Lorentz and Tiersten, [55]. Still other methods of derivation are those in which the governing equations derive from an over-all energy balance law which is postulated to be invariant under Euclidian transformations. The technique, ﬁrst introduced by Green and Rivlin [77] for non-relativistic multipolar theories, was applied to describe deformable polarizable and magnetizable continua by Alblas

3.8 Discussion

87

[9], Parkus [175] and Van de Ven [249]. We have shown in this monograph that all formulations could have been derived from such an over-all energy balance law. When we compare the work of Alblas [9] (who uses the Chu variables) with our model I complete agreement is found. As concerns the work of Parkus [175], he treats magnetizable and polarizable materials separately and when dealing with magnetizable materials restricts himself to quasi-static processes. His treatment of dielectrics on the other hand agrees with that of Toupin. Parkus uses Hamilton’s principle and the same is done by Vlasov and Ishmukhametov [259], who arrive at results which agree with our model III. We could have added other electromagnetic models, if we had wished to do so, and for reasons of completeness these models should also be mentioned. Fano, Chu and Adler [73] and Penfield and `rean descriptions, as do Hutter Haus [177] also present the so called Ampe and Pao [91] in a paper dealing with magnetizable elastic solids with thermal `rean formulation (in which magnetizaand electrical conduction. The Ampe tion is modelled by a relativistic electric circuit) can easily be transformed into the Lorentz- or the statistical formulation and when this is done, it is found that the body force expression agrees with the one presented in model IV. The expressions for the energy supply and the body couple, when restricted to the non-relativistic approximation, fully agree with those of model IV. Maugin on the other hand, partly in collaboration with Eringen and Collet, [131]–[135] uses the principle of virtual power to derive the basic equations for several diﬀerent theories of magnetoelastic interactions. The interest of these authors is generally more limited, as they treat specialized subjects, such as spin relaxation and surface eﬀects, all in the quasi-static approximation. In [45], Collet and Maugin, and in [135], Maugin treat dynamic processes in a non-relativistic approximation. Although there is a slight diﬀerence between our non-relativistic approximation and their owns, which can be traced back to the use of Gaussian units as opposed to MKSA-units, the results of [45] and [135] when brought to our non-relativistic approximation completely agree with model IV. Maugin and Eringen also presented a fully relativistic treatment in [132]. Finally, starting from a four-dimensional ´ [30] derive expressions for relativistic formulation, Boulanger and Mayne the electromagnetic body force and energy supply. In cooperation with Van Geen, they apply their results in [31] for the investigation of magnetooptical, electrooptical and photoelastic eﬀects in elastic polarizable and magnetizable isotropic media. When comparing the relations for momenta and energy following from Eq. (14) of [50] or the balance laws (4)–(6) of [31], no immediate equivalence with one of our models is found. However, their balance laws can easily be related to those of, for instance, model IV by taking in the equations (4)–(6) of [31] ρε = IV (ρU ) + Bi Mi , and

88

3 A Survey of Electromagneto-Mechanical Interaction Models

σij =

IV

tij + Ei Pj + Bi Mj − δij Bk Mk .

(In this context it should be noted that in Eq. (16) of [30] two terms are missing, which should follow from (14), and which re-appear in Eq. (4) of [31].) In conclusion we might justly state that the models I to V embrace within the non-relativistic approximation the description of dynamic theories of polarizable and magnetizable materials known to date. There are formulations simpler than those presented above, but these formulations aim at describing more restrictive situations such as static and quasi-static processes. There are also more complicated descriptions, but those include additional phenomena, as for instance spin interaction, polarization gradient eﬀects and the like. These phenomena are outside the scope of this monograph. Since the completion of this chapter a series of articles did appear which we would have taken into account if being aware of them. To mention are works by Alblas [10], Maugin and Eringen [136], Prechtl [185] and Romano [200]. Alblas presents a general exposition of electro- and magnetoelasticity with special topics such as electro- and magnetostatics including constitutive equations and linearization procedures and magnetoelastic stability. His formulation is that of Chu. Maugin and Eringen formulate a theory of magnetoelastic interaction including electric quadrupoles. Their approach is similar to that of Dixon and Eringen [59] and they seem to correct the erroneous jump condition in this work. When discarding the electric quadrupoles they arrive at body force expressions identical to those of our model IV. Prechtl, on the other hand presents a relativistic Chu formulation; he reduces it to a three dimensional form, which in the non-relativistic approximation is equivalent to our model II. Unfortunately, Prechtl does not present constitutive relations, so that a full comparison is not achieved. Romano gives a quasi-static magnetoelastic Chu formulation.

4 Equivalence of the Models

4.1 Preliminary Remarks In Chap. 3 various descriptions of electromechanical interaction models were laid down, but no attempt was made to search for interrelations amongst these models. In this chapter we perform the ﬁrst steps toward a proof or disproof of the equivalence of these models. Speciﬁcally, we are going to present the conditions that need to be satisﬁed in order to render two formulations equivalent. To begin with, we should point out that two models are called equivalent provided they deliver the same results for physically observable quantities in any arbitrary initial-boundary-value problem. By observable (measurable) quantities we mean all those physical quantities that can be measured uniquely by two diﬀerent independent observers. All kinematical quantities that are derivable from the motion χi (X, t) are measurable in principle. Regarding electromagnetic ﬁeld quantities we take the viewpoint that they are not measurable except in vacuo, where they can be observed by measuring the force exerted on a test charge. Finally, (empirical) temperature is also regarded as being measurable, because, as is known from thermodynamics, it is a measure for the hotness of a body. The instruments to measure temperature are thermometers. They relate our sensation of the hotness of a body to a physical quantity, say pressure or volume, and the latter are measurable. Diﬀerences in the various electromagnetic models appear in the body force, body couple and energy supply of electromagnetic origin, but as is well-known, such diﬀerences may be compensated in the stress tensor, the internal energy and the energy ﬂux. Except for the latter, these quantities are determined by a thermodynamic potential (the Helmholtz free energy or its Legendre transformations). In other words, equivalence of two electromechanical interaction models depends to a large extent on the fact that the free energies of the two formulations can be determined so as to render stress, entropy and electromagnetic ﬁeld variables compatible with the above mentioned compensation between stress and body force, etc. Equivalence of two theories is therefore a thermodynamic requirement. Practically, two formulations are equivalent if upon transforming one into the other, not only the ﬁeld equations but also the initial conditions and jump conditions are alike. Here in this chapter we investigate the equivalence of K. Hutter et al.: Electromagnetic Field Matter Interaction in Thermoelastic Solids and Viscous Fluids, Lect. Notes Phys. 710, 89–102 (2006) c Springer-Verlag Berlin Heidelberg 2006 DOI 10.1007/3-540-37240-7 4

90

4 Equivalence of the Models

the models described in the preceding chapter. In particular, the conditions under which this equivalence is achieved will be formulated. Needless to say that all comparison will be made to within the order of the non-relativistic approximation.

4.2 Comparison of the Models I and II Here we discuss the two-dipole models, and we shall for this purpose heavily rely upon the results presented in Sect. 3.2. To begin with, note that both models are based on the same set of electromagnetic ﬁeld variables and therefore obey the same set of Maxwell equations (3.3.3). Diﬀerences do occur in the electromagnetic body force, body couple and energy supply, which can most easily be identiﬁed by comparing the expressions according to (3.3.7) and (3.3.49). From (3.3.7)1 and (3.3.49)1 we read oﬀ that II

(ρFie ) = I (ρFie ) − (Pj Ei + µ0 Mj Hi ),j .

(4.2.1)

We conclude that the balance laws of momentum (3.3.9)2 and (3.3.50)2 are identical provided that II

tij = I tij + Ei Pj + µ0 Hi Mj .

(4.2.2)

In order to compare the balance laws of moment of momentum and energy, we must compare ρLeij − t[ij] ,

and

ρre + tij x˙ i,j ,

respectively. Using ρLeij as given in (3.3.7)2 and (3.3.49)2 and ρre according to (3.3.12) and (3.3.49)3 , we immediately see that (ρLeij − t[ij] ) =

II

(ρLeij − t[ij] ) ,

(4.2.3)

(ρre + tij x˙ i,j ) =

II

(ρre + tij x˙ i,j ) ,

(4.2.4)

I

and I

provided that the stresses are related by (4.2.2). Hence, with the conditions (4.2.2), the balance laws of model I and model II have been proved to be equivalent. It remains to compare the jump conditions. Since the Maxwell equations are the same, there cannot be a diﬀerence in the jumps of the electromagnetic ﬁeld variables. On the other hand, as can immediately be seen from (3.3.21), (3.3.22) and (3.3.52), the jump conditions for momentum and energy of matter and ﬁelds are identical provided that the stress tensors are related according to (4.2.2).

4.3 Comparison of the Models I and III

91

The above comparison makes no use of thermodynamic arguments. Otherwise stated, equivalence of the two theories is guaranteed only if the relations (4.2.2) hold. That such relations indeed can hold follows from thermodynamic arguments, and they are, although trivial in this case, not obvious in general. In order to invoke these conditions, we must compare theories based on the same constitutive postulates, and, of course, in this case we must choose in both models the same set of independent constitutive variables. Explicitly we have done this for case a): ˆ αβ , Pα , Mα , Θ, Θ,i , Q) , C = C(C and the results are listed as (3.3.32) and (3.3.57). It follows from these that I

tij = 2ρ

and that II

tij = 2ρ

∂ I ψˆ Fiα Fjβ + Pi Ej + µ0 Mi Hj , ∂Cαβ

∂ II ψˆ Fiα Fjβ + 2P(i Ej) + 2µ0 M(i Hj) , ∂Cαβ

and it is a straightforward matter to show that (4.2.2) is satisﬁed if we choose II

ψˆ = I ψˆ ,

(4.2.5)

II

U = IU .

(4.2.6)

or With this choice, all the other dependent constitutive quantities that derive from the free energy, e.g. entropy, electromotive intensity, and magnetomotive intensity are guaranteed to be the same. It thus only remains to mention, that in order to obtain full agreement the energy ﬂux vectors must also be chosen to be the same, i.e. II qi = I qi . (4.2.7) The reader can readily prove the above statements to be correct also for all other constitutive theories that are obtained from the above one by merely interchanging some of the dependent and independent variables. We therefore have proved the Proposition. Within the constitutive class of thermoelastic polarizable and magnetizable materials the two-dipole models I and II are equivalent, provided that the free energies and the energy ﬂux vectors are the same functions of their independent variables.

4.3 Comparison of the Models I and III Before comparing the models I and III one should realize that in model I electromagnetic ﬁelds according to the Chu model are used, whereas in model

92

4 Equivalence of the Models

III the Minkowski-ﬁelds are used. As stated by Penfield and Haus, [177], Ch.7, and as follows from relations (3.3.1) and (3.4.1), these ﬁelds are related in the following way EiM = EiC − µ0 eijk x˙ j MkC ,

DiM = ε0 EiC + PiC ,

HiM = HiC + eijk x˙ j PkC ,

BiM = µ0 HiC + µ0 MiC .

(4.3.1)

According to the deﬁnitions (3.4.3) for PiM and MiM we also have PiM = PiC +

1 eijk x˙ j MkC , c2

MiM = MiC − eijk x˙ j PkC .

(4.3.2)

Hence, to within the non-relativistic approximation, PiC = PiM = Pi ,

and MiC = MiM + eijk x˙ j PkM = Mi ,

(4.3.3)

where both, Pi and Mi are known to be objective under Euclidian transformations. From the above relations it follows immediately that the rest-frame ﬁelds Ei and Hi are equal in both formulations, and, moreover, that polarization and magnetization in the Chu formulation are equal to the corresponding rest-frame ﬁelds in the Minkowski formulation. Then, by comparing (3.3.7)2,3 with the corresponding quantities as given by (3.4.23) and (3.4.24), respectively, it is obvious that the expressions for the body couple and energy supply of model I and model III are identical. Since, in the non-relativistic approximation, in both models the electromagnetic momentum vector gi is zero, it is most convenient to relate the respective electromagnetic body forces by comparing the expressions for the Maxwell stresses tM ij according to (3.3.10)1 and (3.4.25)1 . With the aid of (4.3.1)2,4 , we may in (3.4.25) replace the Minkowski ﬁelds BiM and DiM by the Chu ﬁelds and can then show that to within terms containing a c−2 -factor III M tij

= Ei DjM + Hi BjM − 12 δij (ε0 Ek Ek + µ0 Hk Hk ) = ε0 EiC EjC + µ0 HiC HjC + Ei PjC + µ0 Hi MjC

(4.3.4)

− 21 δij (ε0 EkC EkC + µ0 HkC HkC ) = I tM ij . Hence, we have demonstrated that the balance laws (3.3.17), (3.3.9)2 , (3.3.9)3 and (3.4.22), (3.4.23), (3.4.24), are equivalent, provided that III

tij = I tij ,

III

qi = I qi ,

and

III

U = IU .

(4.3.5)

Under these conditions, and to within the non-relativistic approximation, also the jump conditions for momentum and energy of matter and ﬁelds of model I and model III are equal (cf. (3.3.21), (3.3.22) and (3.4.26)6,7 ). All the above is true, of course, again with the provision that constitutive relations show the dependent constitutive variables to be the same in the

4.4 Comparison of the Models III and IV

93

two theories. This can be seen to be true immediately by comparing the constitutive equations (3.3.38) with those of (3.4.31) (thus I ψ = III ψ). The reader may also show this to be correct, if any one of the dependent and independent variables are interchanged. Thus, we have proved the following Proposition. Within the constitutive class of thermoelastic polarizable and magnetizable materials the two-dipole model I and the MaxwellMinkowski model III are non-relativistically equivalent provided that the constitutive relations for the internal energies (or the free energies) and the energy ﬂux vectors are the same functions of their independent variables.

4.4 Comparison of the Models III and IV We begin with the observation that the Maxwell equations in the statistical formulation IV are written in a form which diﬀers from that in the Minkowski formulation. However, if in the equations (3.4.2) the D M and H M ﬁelds are eliminated by means of (3.4.3) the resulting equations equal (3.5.2). Hence, E, P , B and M are identical in both formulations, and we do not need superscripts to distinguish them. Of course, the same is then true also for the ﬁelds E and M. When with the use of (3.4.13) in the expression for the body force, (3.4.22), Hi and Di are eliminated and in so doing terms of the form ε0 E ×B are neglected (as is justiﬁed in view of the corresponding remarks made in Sect. 1.6) one obtains III

(ρFie ) = QEi + eijk Jj Bk + Pj Ej,i + Mj Bj,i

−µ0 Mj Mj,i + eijk (P j Bk + Pj B k ) ,

(4.4.1)

or III

(ρFie ) =

IV

(ρFie ) − ( 12 µ0 Mj Mj ),i ,

as follows from a comparison with (3.5.3)1 . Hence, the electromagnetic body forces in model III and IV diﬀer in the term ( '1 2 µ0 Mj Mj ,i . However, this term is easy to handle, because substitution of the expression for the body force into the balance laws of momentum (3.4.22) and (3.5.5) reveals that the above term leads to a diﬀerence in the stress tensors III tij and IV tij given by IV

tij =

III

tij − 12 δij µ0 Mk Mk .

(4.4.2)

94

4 Equivalence of the Models

This relation does not change the anti-symmetric part of the stresses, and one can see at once that the expressions for the body couples (3.4.23) and (3.5.3)2 are identical, if the relation (3.4.13) is invoked. As concerns the jump conditions of momentum, it is not diﬃcult to see that (3.4.26)6 is equal to (3.5.9)6 , once Di and Hi are eliminated and, furthermore, equation (4.4.2) is substituted. Because of (4.4.2), it may then be expected that the electromagnetic energy supplies ρre in the models III and IV will also diﬀer. This is indeed the case and as can easily be seen from (3.4.24) and (3.5.3)3 they are related by IV

(ρre ) =

III

(ρre ) − ρ

d d (Bi Mi /ρ) + ρµ0 Mi (Mi /ρ) . dt dt

(4.4.3)

Substituting the expressions (4.4.2) and (4.4.3) into the energy balance law for model IV, (3.5.5)4 , and comparing the resulting equation with (3.4.24), we see that the two energy balance laws are equivalent, if the pertinent internal energies are related by IV

U=

III

U − Bi

III

and provided that the heat ﬂuxes IV

1 Mi Mi Mi + ρµ0 , ρ 2 ρ ρ qi and

qi =

III

IV

(4.4.4)

qi are identical, i.e.

qi .

(4.4.5)

Furthermore, the same holds true for the jump conditions of energy of matter and ﬁelds (3.4.26)7 and (3.5.9)7 as can be proved in the usual way. What remains, is to ﬁnd the conditions for which relations (4.4.2)–(4.4.4) can be made compatible with the constitutive equations. For that purpose we ﬁrst consider the free energies. From equations (4.4.4) and (3.5.11) we obtain IV

ψ= =

IV

U−

III

IV

ηΘ =

III

U−

III

ηΘ − Bi

−1 −1 ψ + Pα Eβ Cαβ − Bα Mβ Cαβ +

1 Mi Mi Mi + ρµ0 ρ 2 ρ ρ

ρ −1 Mα Mβ Cαβ , 2µ0

(4.4.6)

where according to (3.4.29) and (3.4.30) III

˜ αβ , Eα , Mα , Θ) . ψ = ψ(C

(4.4.7)

Here, an important point must be mentioned, namely that the free energy function IV ψ on the far left in (4.4.6) is a function of Cαβ , Pα , Bα and Θ. That this must be so can immediately be seen from (3.5.5)4 and the deﬁnition of IV ψ in (3.5.11). On the other hand, III ψ as given in (4.4.7) is a function of Cαβ , Eα , Mα , Θ, and, similarly, so must be the expression on the right-hand side of (4.4.6). If ψ˜ is used as free energy functional in a constitutive theory of model IV, then instead of (3.5.13) we obtain

4.4 Comparison of the Models III and IV

η=−

∂ ψ˜ , ∂Θ

Pi = −ρ

∂ ψ˜ Fiα , ∂Eα

Hi =

∂ ψ˜ Fiα sgn J , ∂Mα

95

(4.4.8)

and IV

tij = 2ρ

∂ ψ˜ Fiα Fjβ − Ei Pj + µ0 Mi Hj − 12 δij µ0 Mk Mk , ∂Cαβ

in the derivation of which use has also been made of (3.4.13)2 and of the relations ∂ρ ∂ρ Fiα Fjβ = Fjα = −ρδij , (4.4.9) 2 ∂Cαβ ∂Fiα and

−1 ∂Cγδ

∂Cαβ

−1 −1 −1 −1 Fiα Fjβ = − 12 (Fγi Fδj + Fδi Fγj ) .

(4.4.10)

We note that relations (4.4.8) are not only compatible with (3.4.31) but also with (4.4.2); needless to say once more that in all those comparisons the non-relativistic approximation is employed. Moreover, the above comparison has been made for one set of dependent and independent constitutive variables and can of course also be repeated for all other possibilities. In view of the relation (4.4.4) this can be done easily. We shall not repeat the details here and conclude with the Proposition. Within the constitutive class of thermoelastic polarizable and magnetizable materials the Maxwell–Minkowski model III and the statistical model IV can be brought to a one-to-one correspondence provided that the internal energies III U and IV U are related by relation (4.4.4) and, furthermore, that the energy ﬂuxes are the same. There still remains one practical question. Given a free energy function III ψ and another one, IV ψ, each a function of its own variables, how can it be decided that the theories according to the models III and IV are equivalent? To this end, observe that IV ψ as it occurs in (4.4.6) is a function of the variables Cαβ , Pα , Bα and Θ, or IV

+

ψ =ψ (Cαβ , Pα , Bα , Θ) ,

(4.4.11)

whereas III ψ as given by (4.4.7) is a function of Cαβ , Eα , Mα and Θ instead. We now can express IV ψ in terms of the same set of variables as III ψ by substituting into (4.4.11) for Pα the relation Pα = −

∂ ψ˜ Cαβ , ∂Eβ

which follows from (4.4.8)2 and (2.6.7)1 , and for Bα the expression

(4.4.12)

96

4 Equivalence of the Models

Bα =

∂ ψ˜ ρ Cαβ + Mα , ∂Mβ µ0

(4.4.13)

following from (4.4.8)3 , (3.4.13)2 and (2.6.7)2,5 . In this way we obtain + ∂ ψ˜ ∂ ψ˜ ρ IV ψ = F (Cαβ , Eα , Mα , Θ) :=ψ Cαβ , − Cαβ , Cαβ + Mα , Θ . ∂Eβ ∂Mβ µ0 (4.4.14) If this expression is substituted into (4.4.6) and in so doing an identity is obtained, in other words if + ∂ ψ˜ ∂ ψ˜ ρ ψ Cαβ , − Cαβ , Cαβ + Mα , Θ ∂Eβ ∂Mβ µ0 (4.4.15) ∂ ψ˜ ∂ ψ˜ ρ −1 ˜ Eα − Mα − Mα Mβ Cαβ = ψ(Cαβ , Eα , Mα , Θ) − ∂Eα ∂Mα 2µ0 holds as an identity, then one condition that model III is equivalent to model IV is satisﬁed. Of course, the energy ﬂux vectors must also be expressed in the same variables and must also be the same functions of these variables. The above procedure illustrates how, in practice, two theories can be decided to be equivalent. The procedure may in reality be very elaborate, but it shows explicitly that equivalence of theories amounts to comparison of thermodynamic potentials. We have done this here for the sets (Pα , Bα ) in theory IV and (Eα , Mα ) in theory III. In the table below we show which variable sets of these two formulations naturally correspond to each other. The above investigation can be made for each of them, of course.

Model IV Pα , Pα , Eα , Eα ,

Bα Mα Mα Bα

Model Pα , Eα , Pα , Pα ,

III

Mα Hα Hα Mα

4.5 Comparison of the Models IV and V In this section we compare the statistical model IV and the Lorentz model V. With regard to the electromagnetic variables these models diﬀer only in the deﬁnitions of the magnetization vectors, which are related by MiS = MiL − eijk x˙ j PkL = Mi − eijk x˙ j Pk .

(4.5.1)

4.5 Comparison of the Models IV and V

97

Here, as before we have set MiL = Mi ,

and

PiL = Pi .

(4.5.2)

If these relations are used, the Maxwell equations of one formulation transform into those of the other (see (3.5.2) and (3.6.2)). With the transformation rule (4.5.1) it is also straightforward to relate the body forces. This is most easily achieved by comparing the Maxwell stresses (3.5.8)1 and (3.6.8)1 , since in both formulations the electromagnetic momenta are equal to zero. This gives V M tij

=

IV M tij

− EiS Pj + MiS Bj − δij Bk MkS − eikl Pk Bl x˙ j

=

IV M tij

− Ei Pj + Mi Bj − δij Bk Mk .

(4.5.3)

In order for the balance laws of momentum of the two formulations to be equivalent, it is thus necessary that the stress tensors be related by V

tij =

IV

tij + Ei Pj − Mi Bj + δij Bk Mk .

(4.5.4)

This also implies that, if balance of moment of momentum is satisﬁed in model V, in which the stress tensor is symmetric, so it is in model IV. For a comparison of the energy equations, we ﬁrst write (3.5.5)4 in the form IV

(ρU˙ ) =

IV

t(ij) x˙ i,j −

IV

qi,i + Ji Ei + Ei (P i +eijk Mk,j )

(4.5.5)

+(eijk Ej Mk ),i + (Ei Pj − Mi Bj + δij Mk Bk )x˙ (i,j) , for the derivation of which use has also been made of (3.5.6). Comparing (4.5.5) with the balance law of energy of model V, (3.6.6)4 , using thereby relations (4.5.3) and (4.5.4), we infer that the two balance laws are equivalent if V (ρU ) = IV (ρU ) , (4.5.6) and V

qi =

IV

qi − eijk Ej Mk .

(4.5.7)

Note that this relation between the energy ﬂuxes was introduced already in Sect. 3.5 (eq. (3.6.12)), where it was used to bring the entropy inequality into its expected classical form. With relation (4.5.7) the reduced entropy inequalities (3.5.14) and (3.6.19) are then also identical. Bearing the transformation rules (4.5.3), (4.5.4) and (4.5.7) in mind, we further recognize that the jump conditions (3.5.9) and (3.6.9) are also the same. In order to obtain complete equivalence, there remains to consider the constitutive equations. In Chap. 3 the constitutive theories for both models were developed using constitutive assumptions of the form +

C = C (Fiα , Pi /ρ, Bi , Θ, Θ,i , Q) .

98

4 Equivalence of the Models

The results are listed in the two sets of equations (3.5.13) and (3.6.18), and they reveal that relations (4.5.3) and (4.5.4) are satisﬁed, provided that V

+

ψ (Cαβ , Pα , Bα , Θ) =

IV

+

ψ (Cαβ , Pα , Bα , Θ) ,

(4.5.8)

which is in accordance with (4.5.6). Furthermore, this simultaneously guarantees that entropy and electromotive intensity are the same in both formulations. The reader may show himself that these results remain correct when some of the dependent and independent variables are interchanged. We have thus proved Proposition. Within the constitutive class of thermoelastic polarizable and magnetizable materials the non-relativistic statistical model (IV) and the Lorentz model (V) are equivalent provided that the two free energies are the same functions of their variables and the energy ﬂux vectors of the two formulations are related by (4.5.7).

4.6 Conclusions In summary, we have shown that all ﬁeld interaction models presented in Chap. 3 and describing polarizable and magnetizable thermoelastic materials are equivalent to each other and diﬀer only in terms which in the context of the non-relativistic approximation have been considered to be negligibly small anyhow. We may thus justly call these theories to be equivalent. On the other hand, it is true that results obtained for these various models could be diﬀerent in exactly these neglected terms. They have been assumed to be unimportant in all of the above theories, because each of them neglects terms that are preceded by a c−2 -factor. Hence, if experiments for a material obeying our constitutive assumptions should deviate from what any of these formulations predicts, then either the constitutive class must be extended or the non-relativistic approximation must be replaced by a semi-relativistic formulation. There exists a semi-relativistic version of the statistical model (cf. [53]), whereas a semi-relativistic counterpart of the Lorentz model may be derived from a fully relativistic theory of Grot and Eringen [78] by merely neglecting terms of the order of V 2 /c2 . In the same way, still other semi-relativistic formulations can be found in [177], Ch. 7. For all these semi-relativistic formulations the constitutive treatment diﬀers from what we have presented here. Therefore, the equivalence proof of all these models – although claimed to be established by Penfield and Haus,Penfield67 – still remains to be done.

4.6 Conclusions

99

(In [177], neither a constitutive theory nor jump conditions are presented.) This is still one of the important future research topics to be attacked.1 We have performed the equivalence proof for a polarizable and magnetizable solid only, which must be regarded as a severe restriction. There are more complex material behaviors than that dealt with here. For all those this proof is not established yet. One immediate generalization is the inclusion of viscosity in the sense that apart from Fiα its time rate F˙ iα may be included amongst the independent constitutive variables (cf. [9, 134] or [92]). When this is done, it turns out that the stress tensor may be decomposed into two parts: e (4.6.1) tij = tth ij + tij . One part, namely tth ij , is then given by a thermodynamic potential and is ˙ independent of Fiα . The second, dissipative, part teij is called extra stress and is given by an independent constitutive relation involving F˙iα . Since our equivalence proofs have been performed for teij ≡ 0, it immediately follows that models I-V are also equivalent for a viscous polarizable and magnetizable thermoelastic body if, in addition to the conditions stated in this chapter, the extra stress remains the same in all formulations. Another possible extension is obtained when for ferromagnetic materials spin interactions are taken into account. These eﬀects can be reckoned with by including in the constitutive theory the magnetization gradients and by introducing an extra moment of momentum density emanating from the magnetic spin action (see e.g. [9, 249, 233]). This results in two extra terms in the angular momentum law, namely the magnetic spin and the gradient of a magnetic couple stress tensor. The latter is determined by the derivative of the thermodynamic potential (free energy) with respect to magnetization gradients. Moreover, the energy equation must also be supplemented by two 1

On the other hand, if the semi-relativistic formulation is considered to be too complicated, still a small improvement of our non-relativistic approximation can be obtained by changing from SI-units to Gaussian units. In that case, as already said several times before, some c−2 -terms that are neglected in our nonrelativistic approximation become c−1 -terms and must be retained. The most striking eﬀect caused by this change of units, but not the only one, is the fact that the electromagnetic momentum vector g is retained, which becomes (in Gaussian units) 1 (E × H ) , c for the models I, II and III, and 1 (E × B) , c for the models IV and V. If all transformations are executed in a consistent way, all models remain equivalent also in a non-relativistic approximation based on Gaussian units.

100

4 Equivalence of the Models

4.6 Conclusions

Table 4.1. Deﬁnitions and transformations of electromagnetic variables of the ﬁve diﬀerent formulations of electrodynamics and their mutual interrelations

101

102

4 Equivalence of the Models

terms, namely the kinetic energy of the magnetic spin and the energy ﬂux due to the magnetic couple stress. Since these terms always seem to be introduced in a unique way, they do not lead to diﬀerences in the respective formulations. Analogous remarks hold for constitutive theories in which polarization gradients are taken into account (see e.g. [142, 225] or [42]). We conclude this chapter with a Table 4.1 that lists all the pertinent interrelations between the various models. In this table, the relations between the four basic electromagnetic ﬁelds, between the stresses tij , the internal energies U and the energy ﬂuxes qi are listed. The diﬀerences between the latter three quantities are expressed in the ﬁelds Ei , Pi , Bi , Mi , which are identical in all formulations. We did not list the remaining variables, which are also the same in all formulations. They are Q, Ji , ρ, xi = χi (X, t), Θ, η .

5 Material Description

5.1 Motivation In the last two chapters the governing equations of ﬁeld matter interaction were chieﬂy based on the spatial or Eulerian description. Only very brieﬂy the material or Lagrangean formulation was given. Such a formulation is of advantage in describing the deformation of solids, because the boundary conditions for solids are usually prescribed on the undeformed body, which is generally the body in its reference conﬁguration. As a consequence, any theory describing deformable solids in the electromagnetic ﬁelds should from the outset be given in the material description. This is not done in general; on the contrary, in almost all theories the spatial description is applied. The Lagrangean formulation is introduced only afterwards, and if so, only by introducing some approximations, e.g. linearizations. These linearization procedures, although being straightforward are nevertheless quite cumbersome. They become an almost trivial matter when the material description of the governing ﬁeld equations is used from the outset. It thus should be apparent that the Lagrangean formulation is a necessity. The reason for presenting it this late is that it is relatively unknown, so that the equations do at ﬁrst sight not look familiar. They are not new, however, and have been derived before (see Hutter [92, 95]). Use of Lagrangean variables is also made by Alblas, [10], for the linearization of some specialized topics as quasi-electroelastostatics and quasimagnetoelastostatics. This approach can be seen as an improvement of the method used by Toupin, [241]. In his paper on the non-relativistic electrodynamics of deformable media, Prechtl, [185], devoted one section to a Lagrangean formulation of the balance equations and the jump conditions (no constitutive equations are used by him). Both works of Alblas and Prechtl are based on the Chu model. In the following, we shall describe the two-dipole models ﬁrst and shall then pass on to the description of the statistical and the Lorentz formulation; we shall conclude with the Maxwell–Minkowski model.

K. Hutter et al.: Electromagnetic Field Matter Interaction in Thermoelastic Solids and Viscous Fluids, Lect. Notes Phys. 710, 103–146 (2006) c Springer-Verlag Berlin Heidelberg 2006 DOI 10.1007/3-540-37240-7 5

104

5 Material Description

5.2 Material Description of the Two–Dipole Models (Models I and II) In Chap. 2, Sect. 2.3.3, the Lagrangean forms of the Maxwell equations was brieﬂy presented. Those equations appeared in one particular formulation. We could, as was done for the Eulerian description, introduce the various models by simply performing variable transformations. However, little insight into the meaning of the new variables would be gained in doing so, whereas a fresh derivation is enlightening. The Chu formulation is based on the postulations that (i) only two vector quantities are necessary to describe the electromagnetic ﬁelds in free space and, (ii) that material bodies contribute toward these ﬁelds by acting as sources for these ﬁelds. With regard to postulate (i) it is advantageous for the derivation of the Lagrangean–Maxwell equations to work formally with four ﬁeld vectors and to relate the remaining two to the former ones. Concerning postulate (ii) we assume it to be known that magnetization and polarization act as charge and current distributions. Based on such a conception one arrives at the equations (see Hutter [95]) a Bi dν = Qm dν , V

∂V

Ei dli +

d dt

Bia dai = − Jim dai ,

S

∂S

S

(Q + QP )dν ,

Dia dai = V

∂V

Hi dli − ∂S

d dt

d dt

(Ji + JiP )dν ,

Dia dai = S

Qdν + V

(5.2.1)

S

Ji dai = 0 .

∂V

Here, Ei , Hi , Q and Ji are as introduced before; they are the electromotive intensity, the eﬀective magnetic ﬁeld strength and the charge and conductive current densities due to free charges. Moreover, QP , JiP , Qm and Jim are the charge and conductive current densities due to polarization and magnetization, and in the Chu formulation (compare also (3.3.48))

5.2 Material Description of the Two–Dipole Models (Models I and II)

QP dν = − V

Qm dν = −

PiC dai , V

∂V

JiP dai

d = dt

S

PiC dai

µ0 MiC dai ,

∂V

Jim dai

,

S

105

d = dt

(5.2.2) µ0 MiC dai

S

.

S

Finally, we introduced in (5.2.1) the auxiliary ﬁelds Bia and Dia , which are related to the electric and magnetic ﬁeld strength according to Bia = µ0 HiC ,

Dia = ε0 EiC .

(5.2.3)

These are the Maxwell–Lorentz aether relations. To derive a material description from (5.2.1) and (5.2.2) one must by means of relations (2.2.25) simply transform the integrals over spatial volumes and spatial surfaces into integrals over reference volumes and reference surfaces. How this is done is explained in Sect. 2.2.3, and hence we only give results: Bαa dAα = Qm dV , VR

∂VR

∂SR

Bαa dAα = −

SR

Dαa dAα =

d Hα dLα − dt

∂SR

d dt

QdV +

VR

with

SR

Dα dAα =

(J + JP α )dAα , SR

Jα dAα = 0,

Q dV = −

(5.2.4)

∂VR

PC α dAα

P

VR

SR

SR

Jm α dAα ,

(Q + QP )dV , VR

∂VR

d dt

Eα dLα +

VR

SR

Q dV = −

,

∂VR

d JP α dAα = dt

µ0 MC α dAα ,

m

∂VR

PC α dAα ,

Jm α dAα = SR

The following deﬁnitions have been used

d dt

SR

(5.2.5) µ0 MC α dAα .

106

5 Material Description −1 a Bαa = JFαi Bi ,

Bia = J −1 Fiα Bαa ,

−1 Jα = |J|Fαi Ji ,

Ji = |J −1 |Fiα Jα ,

−1 C PC α = |J|Fαi Pi ,

PiC = |J −1 |Fiα PC α ,

Q = |J|Q ,

Q = |J −1 |Q ,

Eα = Fiα Ei ,

−1 Ei = Fαi Eα ,

−1 a Dαa = |J|Fαi Di ,

Dia = |J −1 |Fiα Dαa ,

Hα = Fiα Hi sgnJ ,

−1 Hi = Fαi Hα sgnJ ,

−1 C µ0 MC α = JFαi µ0 Mi ,

µ0 MiC = J −1 Fiα MC α ,

(5.2.6)

some of which were already deﬁned in (2.2.39) but are repeated here for ease of reference. In the above equations integration is over material parts VR , ∂VR , SR and ∂SR , and all variables are thought to be functions of Xα and t so that, as usual, d/dt denotes the time derivative at a ﬁxed particle. The equations (5.2.4) and (5.2.5) may be combined, and for suﬃciently smooth ﬁelds they yield a C + µ0 Mα,α =0, Bα,α

˙ C + eαβγ Eγ,β = 0 , B˙ αa + µ0 M α a + Pα,α = Q , Dα,α

(5.2.7)

−D˙ αa − P˙ α + eαβγ Hγ,β = Jα , ˙ =0. Jα,α + Q Equations (5.2.7) are the Maxwell equations in the material description. We shall call Eα and Hα the Lagrangean electric and magnetic ﬁeld strengths, Pα and MC α the Lagrangean polarization and magnetization of the Chu formulation and Q and Jα the Lagrangean free charge and current densities. Since, in a non-relativistic approximation, the meaning of Pα is unique for all formulations the upper index C for Pα has been dropped. This is not the case for Mα and, therefore, the index C must be retained there. Before we proceed, a few comments seem to be in order: Firstly, all quantities listed on the right column of (5.2.6) are objective vectors (Ji , PiC , Ei , Dia ), objective axial vectors (Bia , MiC , Hi ) and an objective scalar (Q) under the Euclidian transformation group. As a consequence, all Lagrangean variables (listed on the left column of (5.2.6)) are scalars under this group, as they must. Secondly, we can easily relate the variables occurring in (5.2.7) to those listed in (2.2.39), and indeed, this transformation is achieved by setting Bα := Bαa + µ0 MαC ,

Dα := Dαa + Pα ,

(5.2.8)

5.2 Material Description of the Two–Dipole Models (Models I and II)

107

where Dα and Bα may be deﬁned also in terms of Bi and Di as was done in (2.2.39). Thirdly, in order to determine the auxiliary ﬁelds Bαa and Dαa , we simply must use the Maxwell–Lorentz aether relations and must invoke the transformations (5.2.6). When this is done and when terms proportional to c−2 are discarded, a straightforward calculation shows that −1 Dαa = ε0 |J|Cαβ Eβ ,

−1 Bαa = µ0 |J|Cαβ Hβ .

(5.2.9)

These relations thus hold in a non-relativistic theory. When relativistic terms are retained they become much more complicated as can be seen in [95]. Of course, as was the case in the Eulerian description, the global equations (5.2.4) and (5.2.5) also imply jump conditions, which follow from (2.4.5) by invoking the deﬁnitions (5.2.8). They then read a Bα + µ0 MC α Nα = 0 , a Dα + Pα Nα = 0 , eαβγ Eβ Nγ + (Bαa + µ0 MC (5.2.10) α )WN = 0 , a eαβγ Hβ Nγ − (Dα + Pα )WN = 0 , Jα Nα + QWN = 0 . To complete the description of the electro-mechanical interaction the balance laws of mass, momentum, moment of momentum and energy must be given. They are listed in (2.2.27)–(2.2.30), and their local forms appear in (2.2.32). It thus suﬃces to write down the material counterparts of the electromagnetic body force, body couple and energy supply. They are ρ0 Fie , ρ0 Leij and ρ0 re , respectively. To calculate them for models I and II we only need to convert the expressions (3.3.15), (3.3.7)2 , (3.3.12) and (3.3.49) to Lagrangean form omitting thereby c−2 -terms. In this calculation it is also advantageous to use relation (4.2.1): ρ I Fie = ρ II Fie + (Ei Pj + µ0 Hi MjC ),j . The details are tedious even though they are straightforward, and what one obtains reads as follows: For model I I

−1 (Q − Pβ,β )Eα + eαβγ (Jβ + P˙ β )Bγa − µ0 MC (ρ0 Fie ) = Fαi β,β Hα −1 + Fβi (Pα Eβ + µ0 MC , α Hβ ) ,α

−1 (ρ0 Leij ) = F[iα Fβj] (Pα Eβ + µ0 MC α Hβ ) , −1 I e C ˙ ˙ ˙ (ρ0 r ) = Jα Eα + Pα Eα + µ0 Mα Hα + Fαi (Eα Pβ + µ0 Hα MC β )Fiβ , (5.2.11) I

108

5 Material Description

and for model II −1 (ρ0 Fie ) = Fαi {(Q − Pβ,β )Eα + eαβγ (Jβ + P˙ β )Bγa − µ0 MC β,β Hα } , e (ρ0 Lij ) = 0 , II ˙ C Hα , (ρ0 re ) = Jα Eα + P˙ α Eα + µ0 M II II

(5.2.12)

α

in the derivation of which also the identities 1 Fiα eijk Fiα Fjβ Fkγ = Jeαβγ , =0, J ,i Pi,i = |J −1 |Pα,α , P i = |J −1 |Fiα P˙ α ,

˙C, MiC = J −1 Fiα M α

(5.2.13)

C Mi,i = J −1 MC α,α ,

are used. Two conclusions are readily drawn from the above expressions. Firstly, mere inspection of (5.2.11) and (5.2.12) shows that the expressions for body force and body couple are (in a non-relativistic sense) an objective vector and an objective skew-symmetric tensor, respectively. It is also seen that the electromagnetic energy supply is an objective scalar only in model II. Needless to state which model seems to be simpler. Secondly, the body force expression (5.2.12)1 is easily interpretable. It is composed of an electric and a magnetic Lorentz force (compare (3.3.46)1 ) ' e ( −1 II a (Q Eα + eαβγ Jeβ Bγa ) + (Qm Hα + eαβγ Jm (ρ0 Fie ) = Fαi β Dγ ) , (5.2.14) where

Q e = Q − Pα,α ,

Jeβ = J + P˙ β ,

Q m = −µ0 MC α,α ,

˙C Jm β = µ0 Mβ ,

(5.2.15)

as follows from (5.2.2). As was done formally in (2.4.10) we can also derive the Maxwell stress M , the electromagnetic momentum Gi , the energy ﬂux Πα and the tensor Tiα energy density Ω. Formally, this is done by expressing the representations (3.3.10) and (3.3.51) in Lagrangean variables and using the transformations (see (2.4.10)) −1 M = |J|(tM ˙ j )Fαj , Tiα ij − gi x

Gi = |J|gi ,

−1 Πα = |J|(πi − ω x˙ i )Fαi ,

Ω = |J|ω .

(5.2.16)

The calculations are again easy, though rather long, and what one obtains in a non-relativistic formulation can be written as follows:

5.2 Material Description of the Two–Dipole Models (Models I and II)

109

For model II II

−1 −1 M Tiα = Fβi (Dαa Eβ + Bαa Hβ ) − 12 Fαi (Dβa Eβ + Bβa Hβ ) ,

II

Gi = 0 ,

II

−1 Πα = −eαβγ Eβ Hγ − 12 (Dβa Eβ + Bβa Hβ )Fαi x˙ i −1 +(Dαa Eβ + Bαa Hβ )Fβi x˙ i = −eαβγ Eβ Hγ +

II

Ω

(5.2.17) II

M Tiα x˙ i ,

II

Gi = 0 ,

= − 12 (Eα Dαa + Hα Bαa ) .

For model I I

M Tiα =

I

Πα =

−1 M Tiα + Fβi (Pα Eβ + µ0 MC α Hβ ) ,

I

Gi =

−1 Πα + (Pα Eβ + µ0 MC ˙i , α Hβ )Fβi x

I

Ω=

II II

II

(5.2.18) Ω.

The jump conditions for the two models can now be obtained by simply substituting (5.2.17) or (5.2.18) into the general jump conditions (2.4.12). There remains the presentation of the constitutive theory. For that purpose, we ﬁrst eliminate ρ0 rext from (2.3.7) and (2.2.32)3 . When making use of (5.2.14)3 , the following entropy inequality for model II is obtained: ˙ C Hα +Jα Eα − Qα Θ,α ≥ 0 . (5.2.19) −ρ0 ψ˙ −ρ0 η Θ˙ + II Tiα F˙ iα + P˙ α Eα +µ0 M α Θ Here we have introduced the Helmholtz free energy ψ = U − ηΘ ,

(5.2.20)

and have also set

Qα . (5.2.21) Θ The entropy inequality for model I can be found from (5.2.19) by introducing into the latter the following relation between the Piola-Kirchhoff stress tensors of the two models (compare (4.2.2)) Φα =

I

Tiα =

II

−1 Tiα − Fβi (Pα Eβ + µ0 MC α Hβ ) .

(5.2.22)

Inequality (5.2.19) suggests to establish constitutive relations of the form Pα µ0 MC α ˆ , , Θ, Θ,α , Q . (5.2.23) C = C Cαβ , ρ0 ρ0 All variables in the above list are objective per se, although they are diﬀerent from the objective variables used in Chaps. 3 and 4 (say Pα and Mα etc.). That the Lagrangean ﬁeld variables are the natural objective combinations of deformation and electromagnetic ﬁelds will become apparent shortly. However, before deducing constitutive equations, we ﬁrst replace in (5.2.19) the

110

5 Material Description

ﬁrst Piola–Kirchhoff stress tensor Tiα by the second one, which according to (2.2.33) is deﬁned as −1 P := Tiα Fβi . (5.2.24) Tαβ P Once constitutive relations of the form (5.2.23) are postulated for ψ, η, Tαβ , Eα , Hα , Jα and Qα , which all must be objective scalars, we may derive from (5.2.19) in the usual way that i) the free energy cannot depend on Θ,α and Q, Pα µ0 MC α ˆ , ,Θ , (5.2.25) ψ = ψ Cαβ , ρ0 ρ0

and ii) that ∂ ψˆ , ∂Θ ∂ ψˆ Hα = , ∂µ0 MC /ρ0 η=−

∂ ψˆ , ∂Pα /ρ0 ∂ ψˆ II P Tαβ = 2ρ0 , ∂Cαβ

Eα =

(5.2.26)

so that the reduced entropy inequality becomes Jα Eα −

Qα Θ,α ≥0. Θ

(5.2.27)

The second Piola–Kirchhoff stress tensor of model I on the other hand obeys the relation I

P Tαβ = 2ρ0

∂ ψˆ −1 − Cβγ (Pα Eγ + µ0 MC α Hγ ) , ∂Cαβ

(5.2.28)

which, with the aid of (5.2.24) and (5.2.26)4 , can be derived from (5.2.22). P , is not symmetric. However, in Hence, this stress tensor, in contrast to II Tαβ view of (5.2.28) and (5.2.11)2 the balance law of momentum of momentum, which reads (see (2.2.32)) I

P (ρ0 Leij ) = I T[iα Fj]α = −Fiα Fjβ I T[αβ]

(5.2.29)

is satisﬁed identically. When in the energy equation (2.2.32)3 use is made of relations (5.2.20), (5.2.26) and (5.2.12)3 , the latter may be written as ρ0 Θη˙ = Jα Eα − Qα,α + ρ0 rext .

(5.2.30)

It should be noted that this relation is derived for model II, but holds for model I as well. It is this form of the energy equation which normally is used in applications. One advantage of the Lagrangean variables introduced in (5.2.6) is the simplicity which the constitutive equation for stress assumes. Corroboration is provided by a comparison of the formulas (3.3.57)4 and (5.2.26)4 . Another

5.2 Material Description of the Two–Dipole Models (Models I and II)

111

advantage is the form of the Gibbs relation when written in terms of these variables. Indeed, it follows from the deﬁnition of the Helmholtz free energy ψ, (5.2.20), and the results (5.2.26) that ∂η 1 ∂U = , ∂Θ Θ ∂Θ

∂η ∂U 1 = − Eα , ∂Pα /ρ0 Θ ∂Pα /ρ0

∂η ∂U 1 = − Hα , ∂µ0 MC ∂µ0 MC α /ρ0 Θ α /ρ0 ∂U ∂η 1 1 II P = − Tαβ , ∂Cαβ Θ ∂Cαβ 2ρ0

(5.2.31)

from which one readily deduces that

Pα µ0 MC 1 1 II P α Tαβ dCαβ − Eα d − Hα d . (5.2.32) dη = dU − Θ 2ρ0 ρ0 ρ0 Next we would like to explore the consequences, which follow from the Gibbs relation, but were not determined in the previous chapter. The reasons for this postponement are formal ones, for the Gibbs relation assumes a particularly simple form when written in the Lagrangean variables (compare (5.2.32) with (3.3.59)). From a practical point of view, that is from a viewpoint of an applied physicist, who must determine actual constitutive relations by performing appropriate experiments, the Helmholtz free energy is not necessarily the most convenient variable to match experiments with theory. These are rather the internal energy, the stress tensor, the polarization and magnetization per unit mass. In what follows, we shall demonstrate, ﬁrstly, how a consistent constitutive theory can be developed when starting from this end. Secondly, this approach will show that, ultimately one searches for the free energy also when using this more physical approach. The method will further demonstrate that the constitutive relations for internal energy, entropy, free energy and stress can all be separated into two parts, one of which is of purely thermoelastic origin and can therefore be determined from thermoelastic experiments in the absence of electromagnetic ﬁelds. The second parts are then the respective eﬀects due to the electromagnetic ﬁelds. We shall outline the procedure for the case that Pα and Mα are chosen as independent electromagnetic ﬁeld variables. At the center of the following derivation lies the Gibbs relation (5.2.32) or the identities (5.2.31), which are the basis for the derivation of the latter. Note that relations (5.2.31) imply integrability conditions, which can easily be derived by cross diﬀerentiations. If these diﬀerentiations are performed the following chain of identities is obtained: (here Mα ≡ MC α)

112

5 Material Description

P ∂ II Tαβ 1 ∂Θ = = = , ∂U ∂U ∂U Θ P − Eα − Hα 2ρ0 − II Tαβ ∂Pα /ρ0 ∂µ0 Mα /ρ0 ∂Cαβ

−

∂Eα ∂Θ

−

∂Hα ∂Θ

−

∂Eα ∂Hβ ∂Eα ∂Eβ ∂Hα ∂Hβ = , = , = , ∂µ0 Mβ /ρ0 ∂Pα /ρ0 ∂Pβ /ρ0 ∂Pα /ρ0 ∂µ0 Mβ /ρ0 ∂µ0 Mα /ρ0 P P P P ∂ II Tβγ ∂ II Tβγ ∂ II Tαβ ∂ II Tγδ ∂Eα ∂Hα = , 2ρ0 = , = . ∂Cβγ ∂Pα /ρ0 ∂Cβγ ∂µ0 Mα /ρ0 ∂Cγδ ∂Cαβ (5.2.33) It follows that the relations on the right of (5.2.33)1,2,3 are all functions of the temperature alone. All the more, they are exactly 1/Θ and are equal. Such relations, of course, reduce the eﬀective labor of the experimentalist considerably. The next step consists in the integration of the identities (5.2.33). To this end, notice that (5.2.33)1,2 can also be written in the form Eα Hα ∂U ∂U 2 ∂ 2 ∂ = −Θ = −Θ , . (5.2.34) ∂Pα /ρ0 ∂Θ Θ ∂µ0 Mα /ρ0 ∂Θ Θ

2ρ0

As we shall see in a moment, these equations allow us to decompose U into two parts of which one is due to the electromagnetic ﬁelds, whereas the other is the internal energy of the body in the absence of the ﬁelds. To see this, we introduce the 6-tuples xA = (Pα /ρ0 , µ0 Mα /ρ0 ), fA = (Eα , Hα ), (A = 1, 2, . . . , 6; α = 1, 2, 3) . (5.2.35) The two equations (5.2.34) then combine to give ∂U fA 2 ∂ = −Θ A = 1, 2, . . . , 6 . (5.2.36) = FA , ∂xA ∂Θ Θ If the functions fA (or FA ) are known functions of their arguments xA , Cαβ and Θ, equation (5.2.36) is a set of six partial diﬀerential equations for U . Of course, in order that such a system is integrable the functions fA must satisfy integrability conditions, namely ∂fA ∂fB = , ∂xB ∂xA

or

∂FA ∂FB = . ∂xB ∂xA

(5.2.37)

In the above case these are identical with (5.2.33)4,5,6 and consequently, (5.2.36) can indeed be integrated. To construct the solution of (5.2.36) we write it in vector form ∇U = F ,

5.2 Material Description of the Two–Dipole Models (Models I and II)

113

which is more suggestive, because it shows the resemblance with the relation between the potential energy U and a conservative force F , the conservativism of which is assured by (5.2.37). The value of U in a point x can then be found from the line integral x U (x) =

F (x ) · dx + constant ,

0

which is independent of the path transversed from 0 to x. Choosing the straight line from 0 to x, we thus can take x = xs , and then obtain

0≤s≤1,

1 U (x) = x ·

F (xs)ds + constant. 0

Proceeding in an analogous way with (5.2.36), we ﬁnd the following solution: I ∂ (5.2.38) U = U (xA , Cαβ , Θ) = −Θ2 + U 0 (Cαβ , Θ) , ∂Θ Θ where U 0 (Cαβ , Θ) replaces the integration constant in the preceding analysis and I stands for 1 I = I(xA , Cαβ , Θ) = xA

fA (xB s, Cαβ , Θ)ds .

(5.2.39)

0

From this relation it is obvious that I is zero for vanishing electromagnetic ﬁelds (fA = 0). Furthermore, by diﬀerentiating (5.2.39) with respect to xA it immediately follows that ∂I fA = . (5.2.40) ∂xA With the representation (5.2.39) part of our goal is achieved, namely that the internal energy U is separated into two parts. Here, U 0 is the speciﬁc internal energy when the electromagnetic ﬁelds vanish. The term involving I is the correction due to the presence of the electromagnetic ﬁelds. Not all the identities (5.2.33) have been exploited when constructing the solution (5.2.38), (5.2.39). For instance, we still must explore conditions (5.2.33)3 . They can also be written in the form " # II P Tαβ 1 ∂I ∂ 1 ∂U 0 . (5.2.41) = − Θ2 ∂Cαβ ∂Θ Θ ∂Cαβ 2ρ0 Θ

114

5 Material Description

This is simpliﬁed if we introduce the purely thermoelastic part of the second Piola–Kirchhoff stress tensor ◦ II P Tαβ

=

◦ II P Tαβ

◦ II P Tαβ

by

(Cγδ , Θ) =

II

P Tαβ (xA = 0, Cγδ , Θ) .

Then (5.2.41) with I = 0 delivers the relation  ◦

(5.2.42)



II P Tαβ 

0

1 ∂U ∂  =−   , Θ2 ∂Cαβ ∂Θ 2ρ0 Θ whence follows ∂ ∂Θ

◦ 1 ∂I 1 II P II P + Tαβ − Tαβ =0, Θ ∂Cαβ 2ρ0

or after integration II

P Tαβ =

◦ II P Tαβ

+2ρ0

∂I + Θταβ (xA , Cγδ ) . ∂Cαβ

(5.2.43)

With the aid of (5.2.33)7,8 and (5.2.40), ταβ can be shown to be independent of xA , so that ταβ = ταβ (Cγδ ) . Taking this result into account in equation (5.2.43) and evaluating the latter at zero magnetic ﬁelds leads us, with the aid of (5.2.42), to the conclusion that ταβ must be zero. Hence, II

P Tαβ =

◦ II P Tαβ

+2ρ0

∂I , ∂Cαβ

(5.2.44)

an equation which separates the stress into a thermoelastic and a ﬁeld part. Equations (5.2.40) and (5.2.44) may now be used to write the Gibbs relation (5.2.32) in the form

∂I 1 II ◦ P 1 d η+ Tαβ dCαβ , = dU 0 − ∂Θ Θ 2ρ0 whence, since the right-hand side is independent of the electromagnetic ﬁelds, it follows that ∂I . (5.2.45) η = η 0 (Cαβ , Θ) − ∂Θ Thus, it has also been possible to separate the ﬁeld part of the entropy from the corresponding thermoelastic part.

5.2 Material Description of the Two–Dipole Models (Models I and II)

115

After the introduction of ψ 0 = ψ 0 (Cαβ , Θ) , as the free energy in case of vanishing electromagnetic ﬁelds, a nice interpretation of I follows from (5.2.26)1 and (5.2.45). Accordingly, ∂ ˆ ∂ψ 0 (ψ − I) = −η 0 = , ∂Θ ∂Θ

(5.2.46)

and, hence, ˆ A , Cαβ , Θ) = ψ 0 (Cαβ , Θ) + I(xA , Cαβ , Θ) + g(xA , Cαβ ) . ψ(x Use of (5.2.40), (5.2.26)2,3 and the deﬁnition of ψ 0 then shows that g ≡ 0, so that ˆ A , Cαβ , Θ) = ψ 0 (Cαβ , Θ) + I(xA , Cαβ , Θ) . (5.2.47) ψ(x Here, ψ 0 denotes the free energy, when there are no electromagnetic ﬁelds present, and I is therefore the ﬁeld part of the Helmholtz free energy. There is another possible separation of the internal energy, entropy, free energy and the stresses, which, from a practical point of view is as important as the above one. We mean the separation into contributions due to “rigidbody processes” and the corrections due to deformations. This approach gives a natural separation of rigid-body electrodynamics from that of deformable bodies. The idea is to commence the integration of (5.2.33)1,2,3 with (5.2.33)3 , which, with the deﬁnitions yA = (C11 , C22 , C33 , C23 , C31 , C12 ) , P , gA = ( II T11

II

P T22 ,

II

P T33 ,

II

P T23 ,

II

P T31 ,

may be written as ∂ ∂U = −Θ2 ∂yA ∂Θ

II

P T12 ), (A = 1, 2, . . . , 6)

gA 2ρ0 Θ

(5.2.48)

,

(5.2.49)

where in view of (5.2.33) ∂gA ∂gB = , ∂yB ∂yA holds. Using the same approach as before, we derive the following representation: J R 2 ∂ U (xA , yA , Θ) = U (xA , Θ) − Θ , (5.2.50) ∂Θ Θ where yA J = J(xA , yA , Θ) = 2ρ0

1 gA (xB , yB , s, Θ)ds , 0

and, moreover,

(5.2.51)

116

5 Material Description

gA = 2ρ0

∂J , ∂yA

∂J ∂J , , η = η R (xA , Θ) − ∂gxA ∂Θ ˆ A , yA , Θ) = ψ R (xA , Θ) + J(xA , yA , Θ) , ψ(x R (xA , Θ) + fA = fA

(5.2.52)

where we still note that for the derivation of (5.2.52) the Gibbs relation (5.2.32) was written in the form 1 ∂J R dxA ) . = (dU R − fA d η+ ∂Θ Θ In these equations the quantities carrying a superscript R are the internal R , the entropy η R , and the free energy energy U R , the electromagnetic ﬁelds fA R ψ for zero deformation. They constitute the rigid-body contributions and form the constitutive relations of rigid-body electrodynamics. All constitutive properties that can be traced back to deformation are contained in the function J. This completes the constitutive theory. The integration procedure has shown that all diﬃculties rest on an appropriate determination of the functions I or J (or more generally of ψ). Of course, the same procedure can also be taken when some of the dependent and independent variables are interchanged. Since the details of the pertinent calculations are the same as above we leave them to the reader. Before we proceed we would like to mention that there are still further possibilities of separating the various eﬀects. These can easily be derived if either only (5.2.33)1 , or else (5.2.33)2 , is used in the integration process. If the results contained in all these combinations are put together, the free energy can, for instance, be written as a sum of four parts, one of which is the ﬁeld-free energy. The second term involves, apart from the deformation, only the electric eﬀects (polarization); in the third term then only magnetization appears. Only in the fourth term does there occur electromagnetic coupling. Hence, we have shown that the separation of the interaction phenomena into the four physically clearly deﬁned parts is indeed possible – and justiﬁed. For later reference, we would like to list here also the constitutive equations for the case that, instead of Pα /ρ0 and µ0 Mα /ρ0 , Eα and Hα are the independent variables. In other words, we shall postulate constitutive equations of the form ¯ αβ , Eα , Hα , Θ, Θ,α , Q) , (5.2.53) C = C(C which can easily be obtained from the preceding ones by the Legendre transformation µ0 1 ¯ αβ , Eα , Hα , Θ) , (5.2.54) ¯− ηΘ = ψ(C ψ¯ = U − ηΘ − Eα Pα − Hα MC α =ε ρ0 ρ0 which leads to the relations

5.3 Material Description of the Statistical and the Lorentz Formulation

∂ ψ¯ , ∂Eα ∂ ψ¯ = 2ρ0 , ∂Cαβ

η=−

∂ ψ¯ , ∂Θ

Pα = −ρ0

µ0 MC α

∂ ψ¯ = −ρ0 , ∂Hα

II

and to the Gibbs relation

1 1 dη = d¯ ε− Θ 2ρ0

II

P Tαβ dCαβ

P Tαβ

117

(5.2.55)

Pα µ0 Mα + dEα + dHα . ρ0 ρ0

We conclude this section with a few complementary remarks. Firstly, in the above constitutive theory we have introduced the classical expression for the entropy ﬂux, and we have done this already when treating this model in the Eulerian description. That this is correct was proved by Hutter [95], and indeed he obtains exactly the same Gibbs relation. Secondly, taking for ¯ in both models the same function of the form (5.2.25), or (5.2.54), ψ, or ψ, and for the energy ﬂux the same function of the form (5.2.23), or (5.2.53), we automatically guarantee that the two models are thermodynamically equivalent. Finally, we mention that it is particularly easy to linearize the equations in the Lagrangean formulation. We shall demonstrate this in Chap. 6. If we were to follow the order of presentation of the previous chapter, the Lagrangean description of the Maxwell–Minkowski formulation should now follow. For didactic reasons, we shall treat this formulation last.

5.3 Material Description of the Statistical and the Lorentz Formulation (Models IV and V) As we have seen in Chap. 3 already, the statistical formulation and the Lorentz formulation are very similar, and indeed in Chap. 4 we demonstrated how the two models could be brought into a one-to-one correspondence. We therefore expect the two models to show a close interrelation in the Lagrangean formulation as well. This is indeed the case and we shall give corroboration for this below. To begin with, recall that the Maxwell equations in Chap. 2 were stated in terms of the variables Ei , Di , Hi and Bi and that these quantities were related to the statistical and Lorentzian variables in (3.5.1) and (3.6.1), respectively. Using also the relations PiS = PiL = Pi ,

Mi = MiS + eijk x˙ j Pk = MiL ,

(5.3.1)

it is then easy to show that the Maxwell equations (2.2.12), (2.2.13), (2.2.19) and (2.2.20) may be written in the form

118

5 Material Description

d Bi dai + Ei dli = 0 , dt S ∂S Bi dai = 0 , ∂V

−

d dt

Dia dai +

Dia dai ∂V

S

Hia dli =

∂S

Qdν −

= V

Ji dai +

d dt

S

Pi dai +

S

(5.3.2) MiL dli ,

∂S

Pi dai ,

∂V

where we have introduced the auxiliary quantities HIa := HIa − eijk x˙ j Dka ,

Dia := ε0 Ei , with

Hia :=

1 Bi . µ0

(5.3.3)

(5.3.4)

Note that even in a non-relativistic formulation Hia = HIa , but that in this approximation (recall the rules stated just after (3.4.12)) Dia = εEi ,

µ0 Hia = µ0 Hia ,

1 Bi Bi = Hia Bi = Hia Bi . µ0

(5.3.5)

We remind the reader that Ei and Bi are the electric ﬁeld strength and magnetic ﬂux density as they occur in the Minkowski, statistical, and Lorentz formulations. Furthermore, as it was convenient to introduce auxiliary ﬁelds for the two dipole models, so it is here, and the equations (5.3.3)1 and (5.3.4) represent nothing but the familiar Maxwell–Lorentz aether relations. Before we pass on to the presentation of the material description of the equations (5.3.2) a few words are in order regarding the interpretation of the various terms in (5.3.2). Firstly, there is a set of homogenous equations and another one that is inhomogeneous through the presence of free charge and free current terms. As is seen from these, both polarization and magnetization manifest themselves as distributions of surface charges and surface currents, respectively. Secondly, we have expressed the integral laws (5.3.2) in terms of Lorentzian variables, but it is not diﬃcult to write them in terms of the variables of the statistical description by simply invoking the transformations (5.3.1). The only term that changes in this formal substitution is MiL dli = MiS dli + eijk x˙ j Pk dli . (5.3.6) ∂S

∂S

∂S

5.3 Material Description of the Statistical and the Lorentz Formulation

119

It is not convenient to absorb the second member on the right-hand side in the term Hia dai , ∂S

as it occurs on the left-hand side of (5.3.2)3 , because in that case Hia would also be expressed in terms which describe the material behavior. Thus, we conclude that in both the statistical and the Lorentz formulations MiL dli ∂S

is the proper (magnetization) current, and this is one reason for us to regard the Lorentzian description as more advantageous than the statistical description, for it is the former, which will directly lead to a material counterpart of the Eulerian variables. In the statistical formulation this current term must be attributed to magnetization as well as polarization. To arrive at the Lagrangean counterpart of equations (5.3.2) one needs only to transform the integrals back to the reference conﬁguration. If this is done, one obtains d Bα dAα + Eα dLα = 0 , dt S ∂S R R Bα dAα = 0 , ∂VR

−

d dt

Dαa dAα +

Dαa dAα ∂VR

SR

Haα dLα =

∂SR

Q dV −

= VR

Jα dAα + SR

d dt

Pα dAα +

SR

ML α dLα ,

∂SR

Pα dAα ,

∂VR

(5.3.7) and from these one may derive in the usual way the conservation law of charges in the form d Q dV + Jα dAα = 0 . (5.3.8) dt VR

∂VR

All newly introduced variables are already deﬁned in (5.2.6), except for Bα , Haα and ML α , which are given by −1 Bi , Bα = JFαi

Bi = J −1 Fiα Bα ,

Hαa = Fiα Hia sgnJ ,

−1 a Hia = Fαi Hα sgnJ ,

L ML α = Fiα Mi sgnJ ,

−1 L MiL = Fαi Mα sgnJ .

(5.3.9)

120

5 Material Description

C Speciﬁcally, we wish to point out the diﬀerence between ML α and Mα ; these two ﬁelds are related by

ML α =

1 Cαβ MC β . |J|

(5.3.10)

For suﬃciently smooth ﬁelds equations (5.3.7) become Bα,α = 0 , B˙ α + eαβγ Eγ,β = 0 , a = Q − Pα,α , Dα,α

(5.3.11)

−D˙ αa + eαβγ Haγ,β = J + P˙ α + eαβγ ML γ,β , ˙ + Jα,α = 0 . Q These equations are the Maxwell equations in the material description, as they naturally emerge from the Lorentz or the statistical formulations. As was the case in the Chu formulation, they contain two auxiliary variables which can be expressed in terms of Bα and Eα . Indeed, on using relations (5.3.3) and the transformation rules for Ei , Dia , Hia and Bi , a straightforward calculation shows that in a nonrelativistic approximation −1 Dαa = ε0 |J|Cαβ Eβ ,

1 1 a Cαβ Bβ − ε0 eµβγ Cαβ Fjγ x˙ j Eµ . Hα = |J| µ0

(5.3.12)

Recognize, as was already the case in the material description of the two dipole models, that it is through the Maxwell-Lorentz aether relations that the formal linearity of the equations (5.3.11) is destroyed. Nevertheless, equations (5.3.11) appear in a form, which is identical to that in the statistical description when spatial coordinates are used (see (3.5.2)). Variables are, however, diﬀerent and so are the conﬁgurations. One could, if one so desired, write (5.3.11) also as Bα,α = 0 , B˙ α + eαβγ Eγ,β = 0 , Dα,α = Q , −D˙ α + eαβγ Hγ,β = Jα , ˙ + Jα,α = 0 . Q where

(5.3.13)

5.3 Material Description of the Statistical and the Lorentz Formulation

Dα := Dαa + Pα

and

Hα := Hαa − ML α ,

121

(5.3.14)

and would in this way formally arrive at a material Minkowski formulation. We shall come back to this in the next section. Note that (5.3.13) agrees with (2.2.40) and thus Dα and Hα also agree with Dα and Hα introduced there. The balance laws (5.3.2) imply also jump conditions, which can easily be derived from (2.4.5) by simply invoking (5.3.14). This yields Bα Nα = 0 , eαβγ Eβ Nγ + Bα WN = 0 , a Dα + Pα Nα = 0 , (5.3.15) a a eαβγ Hβ − ML β Nγ − (Dα + Pα )WN = 0 , Jα Nα + QWN = 0 . To complete the description, we must also derive the Lagrangean versions of the body force, body couple and energy supply expressions of electromagnetic origin. For this purpose we simply express these quantities, which in (3.5.3) and (3.6.5) are written in terms of the Eulerian variables, in the Lagrangean ﬁelds Eα , Bα , Pα , ML α etc. Starting from (3.6.5), we obtain for the Lorentz formulation (model V) −1 V (Q − Pβ,β )Eα + eαβγ (Jβ + P˙ β )Bγ (ρ0 Fie ) = Fαi L , +(ML α,β − Mβ,α )Bβ (5.3.16) V (ρ0 Leij ) = 0 , (ρ0 re ) = Jα Eα + P˙ α Eα + eαβγ ML γ,β Eα .

V

For the derivation of the corresponding expressions of the statistical model it is convenient to make use of the relation (compare (4.5.4)) IV

(ρFie ) =

V

(ρ0 Fie ) + (Ei Pi − Mi Bj + δij Mk Bk ),j ,

(5.3.17)

an expression, which can be derived from (3.5.3) and (3.6.5). Alternatively, the expressions for the body couple and energy supply follow from (3.5.3)2 and (3.5.3)3 , respectively, the latter written in the form

(ρre ) = Ji Ei + (P i +Pj x˙ i,j )Ei − (B i −Bi x˙ j,j + Bj x˙ i,j )Mi . (5.3.18)

IV

Transforming the above expressions to referential coordinates, we obtain for model IV

122

5 Material Description

−1 (Q − Pβ,β )Eα + eαβγ (Jβ + P˙ β )Bγ (ρ0 Fie ) = Fαi L +(ML − M )B β α,β β,α ( ' −1 L + Fαi (Eα Pβ − ML α Bβ + δαβ Mγ Bγ ) ,β ,

IV

−1 (ρ0 Leij ) = F[iα Fβj] (Pα Eβ − Bα ML β) ,

IV

(5.3.19)

˙ (ρ0 re ) = Jα Eα + P˙ α Eα − ML α Bα −1 −1 L . +F˙kγ (Eβ Pγ − ML β Bγ )Fβk + Mβ Bβ Fγk

IV

We would like to point out that the body force expression (5.3.16)1 is seemingly diﬀerent from that given in [92]. However, when using the identity −1 −1 −1 −1 −1 −1 L L Fνi,µ Bµ ML ν − Fkδ Fµi Fνk,µ Bδ Mν = Fkµ (Fνi,k − Fνk,i )Bµ Mν = 0 , (5.3.20)

which was not observed in [92], the body force expressions turn out to be identical. We see that the body force expressions in the two formulations diﬀer by a term that is the divergence of a tensor. It corresponds to the divergence term occurring already in the Eulerian body force. Particularly interesting is a comparison of the electromagnetic energy supply terms listed in (5.3.16)3 and (5.3.19)3 . Using (5.3.11)2 , we can show that IV(ρ0 re ) may also be written as L (ρ0 re ) = Jα Eα + P˙ α Eα + eαβγ ML γ,β Eα + (eαβγ Eβ Mγ ),α −1 −1 L +F˙ kγ (Eβ Pγ − ML . β Bγ )Fβk + Mβ Bβ Fγk

IV

(5.3.21)

Recognizing that, as already said several times before, for a comparison of the energy supplies of two formulations we need not compare the term ρ0 re alone but rather the combination (ρ0 re + tij x˙ i,j ). Thus when performing this comparison we see that the last term of expression (5.3.21) for IV(ρ0 re ) originates from the diﬀerence in the stress tensors of the models IV and V. Indeed with the aid of (4.5.4) we may easily show that IV

Tiα =

V

−1 L L −1 Tiα + Fβi (Pα Eβ − Bα ML β ) − Mγ Mγ Fαi .

(5.3.22)

However, the term (eαβγ Eβ ML γ ),α cannot be explained in this way. Consequently either the energy ﬂuxes Qα or the entropy ﬂuxes φα of the two formulations must diﬀer. Indeed, we may choose, as was already done in the Eulerian description, L QSα = QL α + eαβγ Eβ Mγ ,

(5.3.23)

5.3 Material Description of the Statistical and the Lorentz Formulation

123

and then obtain L QL QSα α + eαβγ Eβ Mγ = = φL (5.3.24) α . Θ Θ This form of the entropy ﬂux vector was proved to be correct by Hutter [92] in a theory of viscous isotropic thermoelastic, polarizable and magnetizable solids. Here we treat (5.3.23) as a postulate. When we compare the expressions for the electromagnetic energy supply (5.3.16)3 and (5.3.19)3 with the corresponding Eulerian expressions, (3.5.3)3 and (3.6.5)3 , we recognize that the Lorentzian expression V(ρ0 re ) is formally the same as its spatial counterpart V(ρre ). This property is not shared by IV (ρ0 re ) and IV(ρre ). Finally, we note that the expressions for the electromagnetic body couple and body force are simpler in the Lorentz formulation than in the statistical formulation. All this, of course, are reasons which make the Lorentz formulation to be (formally) superior to the statistical one, although, as we have shown, they are equivalent. We proceed with the jump conditions for momentum and energy of matter and ﬁelds as they are listed in their general form in (2.4.12). Therefore, M , Gi , Πα and Ω. These quantities are obtained if it suﬃces to evaluate Tiα use is made of the transformations (5.2.16) for tM ij , gi , πi and ω, which in (3.5.8) and (3.6.8) are given for the statistical and the Lorentz formulations, respectively. When the indicated transformations are performed the following relations are obtained:

φSα =

(i) in the Lorentz formulation (model V) V

−1 −1 M Tiα = Fβi (Dαa Eβ + Bα Hβa ) − 12 Fαi (Dβa Eβ + Bβ Hβa ) ,

V

Gi = 0 ,

V

−1 −1 Πα = −eαβγ Eβ Hγa + Fβi (Dαa Eβ + Bα Hβa )x˙ i − 12 Fαi (Dβa Eβ + Bβ Hβa )x˙ i

= −eαβγ Eβ Hγa + V

Ω

V

M Tiα x˙ i ,

= − 12 (Dαa Eα + Bα Hαa ) , (5.3.25)

(ii) in the statistical formulation (model IV) V

−1 M L Tiα + Fβi (Pα Eβ − Bα ML β + δαβ Bγ Mγ ) ,

IV

M Tiα =

IV

Gi = 0 ,

IV

Πα =

V

−1 L L Πα + eαβγ Eβ ML ˙i γ + Fβi (Pα Eβ − Bα Mβ + δαβ Bγ Mγ )x

= −eαβγ Eβ (Hγa − ML γ)+ IV

Ω

=

V

IV

M Tiα x˙ i ,

Ω. (5.3.26)

124

5 Material Description

When we substitute the above expressions into (2.4.12), what emerges are the jump conditions for momentum and energy of matter and ﬁelds. They will not be written down explicitly. There remains to formulate the constitutive theory of the two models. To this end we derive the reduced entropy inequality for model V by eliminating ρ0 rext from the energy equation and the entropy inequality, taking thereby into account that ρ0 re is given by (5.3.16)3 and the entropy ﬂux by (5.3.24), respectively. In this way we obtain − ρ0 ψ˙ − ρ0 η Θ˙ +

V

˙ Tiα F˙ iα + Eα P˙ α − ML α Bα + Eα Jα −

QSα Θ,α ≥ 0 , (5.3.27) Θ

where V Tiα is the ﬁrst Piola-Kirchhoff stress tensor in the Lorentz formulation. It should be noticed that in (5.3.27) the heat ﬂux vector QSα is used instead of QL α. Constitutive relations are now written as +

C = C (Cαβ , Pα /ρ0 , Bα , Θ, Θ,α , Q) .

(5.3.28)

A short calculation then shows, as usual, that the Helmholtz free energy is independent of Θ,α and Q, +

ψ = U − ηΘ = ψ (Cαβ , Pα /ρ0 , Bα , Θ) , and that

+

(5.3.29)

+

∂ψ η=− , ∂Θ

∂ψ Eα = , ∂(Pα /ρ0 ) +

∂ψ ML , α = −ρ0 ∂Bα

+ V

P Tαβ

(5.3.30)

∂ψ = 2ρ0 . ∂Cαβ

P Here, V Tαβ is the second Piola-Kirchhoff stress tensor, which is deﬁned in terms of Tiα in (2.2.33). With these results the energy equation reduces to its ultimate form

ρ0 Θη˙ = Eα Jα − QSα,α + ρ0 rext .

(5.3.31)

On the other hand, the reduced entropy inequality takes the form Eα Jα −

QSα Θ,α ≥0, Θ

and the Gibbs relation reads

Pα 1 ML 1 V P Tαβ dCαβ − Eα d dη = + α dBα , dU − Θ 2ρ0 ρ0 ρ0 an equation that was also derived in [92].

(5.3.32)

(5.3.33)

5.4 Material Description of the Maxwell–Minkowski Formulation

125

The results for model IV and V are identical, except for the stress tensors of which the relation is given in (5.3.22). It follows from the latter and (5.3.30)4 that + IV

P Tαβ

∂ψ −1 L L −1 = 2ρ0 + (Pα Eγ − Bα ML γ )Cβγ − Mγ Mγ Cαβ . (5.3.34) ∂Cαβ

With this constitutive relation for the second Piola-Kirchhoff stress tensor, the balance law of angular momentum for the statistical formulation is satisﬁed identically. For later use, we wish to give constitutive relations also of the form ˇ αβ , Eα , Bα , Θ, Θ,α , Q) . C = C(C

(5.3.35)

With the Legendre transformation 1 ˇ αβ , Eα , Bα , Θ) , ψˇ = U − ηΘ − Eα Pα = εˇ − ηΘ = ψ(C ρ0

(5.3.36)

this immediately leads to ∂ ψˇ , ∂Eα ∂ ψˇ ∂ ψˇ V P ML , Tαβ = 2ρ0 , α = −ρ0 ∂Bα ∂Cαβ

1 Pα ML 1 V P dη = Tαβ dCαβ + dEα + α dBα . dˇ ε− Θ 2ρ0 ρ0 ρ0 η=−

∂ ψˇ , ∂Θ

Pα = −ρ0

(5.3.37)

As was done in the Chu formulation we could now, if we so desired, also explore the consequences implied by the Gibbs relation and would then again be able to show that constitutive relations and free energy are separable into a thermoelastic and a ﬁeld part or else, a rigid-body part and a part due to deformation. The procedure is analogous to that shown before, and therefore we leave the pertinent details to the reader.

5.4 Material Description of the Maxwell–Minkowski Formulation We saw in Chap. 3, Sects. 3.4 & 3.5 that the spatial electromagnetic ﬁeld variables in the Maxwell–Minkowski formulation are formally the same as those in the statistical description, and that the latter are closely related to the variables in the Lorentz formulation. We therefore adopt the same material electromagnetic variables as in Sect. 5.3 of this Chapter, which are given by (5.3.9). There is only one diﬀerence that must be noted. Magnetization and polarization are regarded as auxiliary variables, whilst the dielectric

126

5 Material Description

displacement and the magnetic ﬁeld strength are considered basic. The Lagrangean counterparts of these ﬁelds can easily be read oﬀ from (5.3.14), and it is not diﬃcult to show that Dα and Hα are related to Hi and Di according to −1 Di , Dα = |J|Fαi

Hα = Fiα Hi sgnJ ,

1 Fiα Dα , |J| −1 Hi = Fαi Hα sgnJ . Di =

(5.4.1)

The Maxwell equations, expressed in Eα , Bα , Dα and Hα are already given in (5.3.13), and we refrain from repeating them here. Before passing on to the presentation of electromagnetic body force, body couple and energy supply, one remark concerning the auxiliary ﬁelds must be made. In view of the properties of this formulation as just outlined one would expect Pα and ML α to be the adequate auxiliary variables. However, since the description becomes formally much simpler when MC α is used instead, we shall prefer to use the latter. In that case the auxiliary ﬁelds may be obtained from (5.3.14), (5.3.10) and (5.3.12); they are −1 Pα = Dα − ε0 |J|Cαβ Eβ , −1 µ0 MC α = Bα − µ0 |J|Cαβ Hβ .

(5.4.2)

In Sect. 3.4 the expressions for the electromagnetic body force, body couple and energy supply are derived directly from the global energy balance law (3.4.19). When written in material form the latter reads d 1 ρ0 U + 2 ρ0 x˙ i x˙ i dV − ρ0 rext + ρ0 Fiext x˙ i dV dt VR VR − {Tiα x˙ i − Qα }dAα d = dt

∂VR −1 − 12 |J|Cαβ (ε0 Eα Eβ + µ0 Hα Hβ ) dV

VR

(5.4.3)

−1 −eαβγ Eβ Hγ + Fβi (Dα Eβ + Bα Hβ )x˙ i + ∂VR −1 −1 − 12 |J|Fαi Cβγ (ε0 Eβ Eγ + µ0 Hβ Hγ )x˙ i dAα .

From this equation the expressions for III Ω and III Πα can directly be read oﬀ. They are equal to the integrands of the ﬁrst and the second integral on the right-hand side of (5.4.3). With the aid of the Maxwell equations (5.3.13) and relations (5.4.2) this balance law can be transformed into the form

5.4 Material Description of the Maxwell–Minkowski Formulation

127

−1 [ρ0 x ¨i − ρ0 Fiext − Tiα,α − Fβi (Dα Eβ + Bα Hβ ) VR

−1 −1 Cβγ (ε0 Eβ Eγ + µ0 Hβ Hγ ) ]x˙ i + ρ0 U˙ − 12 |J|Fαi ,α

(5.4.4)

˙ C Hα −ρ0 r + Qα,α − Jα Eα − P˙ α Eα − µ0 M α −1 ˙ −[Tiα + Fβi (Pα Eβ + µ0 MC α Hβ )]Fiα dV . ext

By applying invariance requirements as was done in Sect. 2.3, it is possible to derive from (5.4.4) local balance equations of linear and angular momentum and energy in a material formulation. Comparison of these equations with those given in (2.2.32) then yields the material versions of the electromagnetic body force, body couple and energy supply. In the derivation of the angular ˙ C are objective momentum equation it must be observed that P˙ α and M α quantities under the Euclidian transformation group (in contrast to P˙i and M˙ i ) this because Pα and MC α are objective scalars. In this way we obtain −1 III (ρ0 Fie ) = Fβi (Dα Eβ + Bα Hβ ) −1 −1 − 12 |J|Fαi Cβγ (ε0 Eβ Eγ + µ0 Hβ Hγ )

,α

−1 QEα + eαβγ Jβ Bγ + Pβ Eβ,α + µ0 MC = Fαi β Hβ,α

(5.4.5)

−1 +eαβγ (Dβ B˙ γ + D˙ β Bγ ) + Fβi,j Fjα (Pα Eβ + µ0 MC α Hβ ) , −1 (ρ0 Leij ) = F[iα Fβj] (Pα Eβ + µ0 MC α Hβ ) ,

III

˙ C Hα + F −1 (Pα Eβ + µ0 MC Hβ )F˙iα . (ρ0 re ) = Jα Eα + P˙ α Eα + µ0 M α α βi

III

Of course, these expressions can also be obtained from their spatial versions (3.4.22), (3.4.23) and (3.4.24) by transforming the latter into their Lagrangean counterparts. Furthermore, from (5.4.5)1 and (5.4.3) the electromagnetic momentum M , electromagnetic energy ﬂux Πα , and energy density Gi , Maxwell stress Tiα Ω are obtained as

128

5 Material Description

III

−1 −1 −1 M Tiα = Fβi (Dα Eβ + Bα Hβ ) − 12 |J|Fαi Cβγ (ε0 Eβ Eγ + µ0 Hβ Hγ ) ,

III

Gi = 0 ,

III

−1 Πα = −eαβγ Eβ Hγ + Fβi (Dα Eβ + Bα Hβ )x˙ i −1 −1 − 12 |J|Fαi Cβγ (ε0 Eβ Eγ + µ0 Hβ Hγ )x˙ i

= −eαβγ Eβ Hγ + III

Ω

III

(5.4.6)

M Tiα x˙ i ,

−1 = − 21 |J|Cαβ (ε0 Eα Eβ + µ0 Hα Hβ ) .

Substitution of (5.4.6) into (2.4.12) then yields the jump conditions for momentum and energy of matter and ﬁelds. There remains to formulate the constitutive theory. To this end, the reduced entropy inequality must be derived. Introducing ψ = U − ηΘ − and φα =

1 Eα Pα , ρ0

Qα , Θ

(5.4.7)

(5.4.8)

we obtain, as usual, ˙ C Hα −ρ0 ψ˙ − ρ0 η Θ˙ − Pα E˙ α + µ0 M α Qα Θ,α −1 ˙ ≥0. +[Tiα + Fβi (Pα Eβ + µ0 MC α Hβ )]Fiα + Jα Eα − Θ Assuming constitutive relations of the form C ˜ Cαβ , Eα , µ0 Mα , Θ, Θ,α , Q , C=C ρ0 µ0 MC α ψ = ψ˜ Cαβ , Eα , ,Θ , ρ0 ∂ ψ˜ ∂ ψ˜ ∂ ψ˜ , Pα = − , Hα = , η=− ∂Θ ∂Eα ∂µ0 MC α /ρ0 ∂ ψ˜ −1 III Tiα = 2ρ0 Fiβ − Fβi (Pα Eβ + µ0 MC α Hβ ) . ∂Cαβ

(5.4.9)

(5.4.10)

we can show that

(5.4.11)

(5.4.12)

Moreover, it is easily shown that with the relation (5.4.12)4 the balance law of moment of momentum is satisﬁed identically. Furthermore, III

P Tαβ = 2ρ0

∂ ψ˜ −1 − Cβγ (Pα Eγ + µ0 MC α Hγ ) . ∂Cαβ

(5.4.13)

When use is made of (5.4.12), (5.4.9) reduces to the residual inequality

5.4 Material Description of the Maxwell–Minkowski Formulation

Eα Jα −

129

Qα Θ,α ≥0. Θ

On the other hand, the energy balance reduces in form to relation (5.2.30), and (5.4.7) and (5.4.12) imply ∂η 1 ∂ ε˜ ∂η 1 ∂ ε˜ = , = + Pα , ∂Θ Θ ∂Θ ∂Eα Θ ∂Eα ∂ ε˜ ∂η 1 = − Hα , (5.4.14) C ∂µ0 MC /ρ Θ ∂µ M 0 0 α /ρ0 α

∂ ε˜ 1 1 III P ∂η −1 C = − Tαβ + Cβγ (Pα Eγ + µ0 Mα Hγ ) , ∂Cαβ Θ ∂Cαβ 2ρ0 where ε˜ := U −

1 Eα Pα = ψ˜ − ηΘ , ρ0

from which the Gibbs relation

µ0 MC 1 α dη = d˜ ε + Pα dEα − Hα d Θ ρ0 1 III P −1 − Tαβ + Cβγ (Pα Eγ + µ0 MC H ) dC , αβ α γ 2ρ0

(5.4.15)

(5.4.16)

may be derived. When the functional ψ as deﬁned by (5.4.7) is replaced by the Helmholtz free energy ˆ αβ , Pα /ρ0 , µ0 MC /ρ0 , Θ) , (5.4.17) ψ = U − ηΘ = ψ(C α ˆ and the constitutive equations (5.4.12) only change in that ψ˜ becomes ψ, that the second equation must be replaced by Eα =

∂ ψˆ . ∂Pα /ρ0

(5.4.18)

Comparison of the results of this section with those of Sect. 4.2 immediately shows that the Lagrangean formulations of models I and III are equivalent. As we have already proved the thermodynamic equivalence of models I and II in Sect. 4.2 and of the models IV and V in Sect. 4.3, we need for a comparison of the Lagrangean formulations of the various models only consider one model out of each of the following groups: (i) models I, II and III, (ii) models IV and V. This comparison between models II and V will be made in Sect. 5.6.

130

5 Material Description

5.5 Thermostatic Equilibrium – Constitutive Relations for Energy Flux and Electric Current In the above, we derived constitutive equations for entropy, stress and two electromagnetic ﬁeld vectors, all of which turned out to be derivable from a free energy. We did not present constitutive relations for the free current Jα and the energy ﬂux vector Qα . These must be given separately and they are restricted by the residual inequality γ := Eα Jα −

Qα Θ,α ≥0. Θ

(5.5.1)

Here, Qα denotes the heat ﬂux in all but the Lorentz formulation, where it must be replaced by the right-hand side of (5.3.23). As was the case in the Eulerian description, exploitation of (5.5.1) depends on whether we are dealing with an electrical conductor or insulator. Hence, we shall discuss the two cases separately. (a) For an electrical insulator (Jα = 0) the residual inequality (5.5.1) reduces to Qα Θ,α ≥0. (5.5.2) γ=− Θ In thermostatic equilibrium, which will again be characterized by the & index (·)&E , that is for time-independent processes with uniform temperature, & γ &E = 0 . Since γ is non-negative in general, it thus assumes its minimum for thermostatic equilibrium. Of necessity then & & & ∂γ && ∂2γ & = 0 , is positive-semi deﬁnite , (5.5.3) & ∂Θ,α E ∂Θ,α ∂Θ,β &E or, when expressed in terms of Qα , & & ∂Q(α && is negative-semi deﬁnite . Qα &E = 0 , ∂Θ,β) &E

(5.5.4)

To see what consequences these relations impose on Qα we consider the case in which constitutive relations are prescribed in the form ˇ αβ , Eα , Bα , Θ, Θ,α , Q) . C = C(C

(5.5.5)

A necessary and suﬃcient condition for the energy ﬂux vector Qα to vanish in thermostatic equilibrium is to write Qα = −ˇ καβ (Cγδ , Eγ , Bγ , Θ, Θ,γ , Q)Θ,β ,

(5.5.6)

5.5 Thermostatic Equilibrium – Constitutive Relations

131

& and this implies that κ ˇ (αβ)&E must be positive-semi deﬁnite. Nothing can & be said about the skew-symmetric part κ ˇ [αβ]&E , but when one restricts oneself to small deformations and small deviations from thermostatic equilibrium it follows from the Onsager relations that κ ˇ [αβ] (δγδ , Eγ , Bγ , Θ, 0, Q) = 0 .

(5.5.7)

(b) In an electrical conductor we may write ˇ α (Cβγ , Eβ , Bβ , Θ, Θ,β , Q) , Qα = Q Jα = ˇ Jα (Cβγ , Eβ , Bβ , Θ, Θ,β , Q) .

(5.5.8)

Thermostatic equilibrium is deﬁned here as a time-independent process with uniform temperature and vanishing electric ﬁeld strength Eα . Hence, & γ &E = 0 , must hold here too, and thus the following conditions emerge & & ∂γ && ∂γ && = 0 , and =0, ∂Eα &E ∂Θ,α &E 

∂2γ  ∂Eα ∂Eβ   ∂2γ ∂Eα ∂Θ,β

 ∂2γ ∂Eα ∂Θ,β    ∂2γ & ∂Θ,α ∂Θ,β &

(5.5.9) is positive-semi deﬁnite. E

Of necessity then & Jα &E = 0 , as well as  ∂ˇ J(α  ∂E  β)  ˇ ˇβ  ∂ Jα 1 ∂Q − ∂Θ,β Θ ∂Eα

and

& Qα &E = 0 ,

 ˇβ 1 ∂Q ∂ˇ Jα − ∂Θ, β Θ ∂Eα    ˇ (α  1 ∂Q − & Θ ∂Θ,β) &

(5.5.10)

is positive-semi deﬁnite . E

(5.5.11) These conditions are satisﬁed provided that (Q)

Qα = −καβ Θ,β + βαβ Eβ , (J)

Jα = βαβ Θ,β + σαβ Eβ , and that

(5.5.12)

132

5 Material Description

  

(J)

βαβ −

σ(αβ) (J) βαβ

1 (Q) − ββα Θ

1 (Q)  β Θ βα  

1 κ(αβ) Θ

is positive-semi deﬁnite . & &

E

(5.5.13) Necessary conditions for (5.5.13) to be satisﬁed are that σ(αβ) and κ(αβ) are positive deﬁnite, but these conditions are by no means suﬃcient. Conditions of suﬃciency include for instance also & & 1 (Q) & (J) & (5.5.14) σ[αβ] &E = 0 , κ[αβ] &E = 0 , and βαβ &E = ββα &E . Θ These represent the well-known Onsager relations.

5.6 Recapitulation and Comparison In the preceding sections we presented the material versions of the two dipole models, of the statistical and Lorentz formulations and of the MaxwellMinkowski formulation. The basic principle in this derivation consisted in transforming known equations into the Lagrangean form by introducing new variables which are particularly convenient in this material description. Since this chapter is heavily loaded with partly complicated formulas, it might be advantageous when the basic ideas are recollected. Key idea behind the Lagrangean description is to derive the equations of electromechanical interactions in a form, which can directly be used in the theory of solid bodies. All equations should therefore be referred to the reference conﬁguration. While such a formulation is well-known in continuum mechanics, it is hardly used in electrodynamics. This is the reason why most theories of solids of electromechanical interactions are treated in the spatial description. Whereas the advantages of the material description will be described extensively in Chap. 6, we would like to draw the reader’s attention here to formal diﬀerences and similarities of the various formulations only. To this end, consider the Maxwell equations ﬁrst. They are listed in (5.2.8) and (5.3.11) for the two-dipole and the Lorentz or statistical models, respectively. For clarity of presentation they will be repeated here. In the Chu formulation they read a C = −µ0 Mα,α , Bα,α

˙ C, B˙ αa + eαβγ Eγ,β = −µ0 M α a C = Q − Pα,α , Dα,α

−D˙ αa + eαβγ Hγ,β = Jα + P˙ αC , ˙ + Jα,α = 0 , Q

(5.6.1)

5.6 Recapitulation and Comparison

133

whereas in the statistical and Lorentz formulations they are Bα,α = 0 , B˙ α + eαβγ Eγ,β = 0 , a = Q − PL Dα,α α,α ,

(5.6.2)

a L −D˙ αa + eαβγ Hγ,β = Jα + P˙ L α − eαβγ Mγ,β ,

˙ + Jα,α = 0 . Q Both sets of equations allow the presentation of the Lagrangean version of the Minkowski formulation by introducing either Bα = Bαa + µ0 MαC ,

Dα = Dαa + PαC ,

(5.6.3)

Hα = Hαa − ML α ,

Dα = Dαa + PL α .

(5.6.4)

−1 L and MαC = |J|Cαβ Mβ

(5.6.5)

or This then yields PαC = PL α = Pα ,

so that the Lagrangean form of the Maxwell equations in the Minkowski formulation becomes Bα,α = 0 , B˙ α + eαβγ Eγ,β = 0 , Dα,α = Q ,

(5.6.6)

−D˙ α + eαβγ Hγ,β = Jα , ˙ + Jα,α = 0 . Q Equations (5.6.1), (5.6.2) and (5.6.6) are the only ones that emerge in the Lagrangean description from all presently known electromagnetic formulations. We have presented in Chap. 3 four diﬀerent formulations, but there are `re formulation of Penfield and Haus [91]) others (for instance the Ampe and all these reduce to (5.6.1), (5.6.2) or (5.6.6). The Lagrangean description has therefore reduced this number to at most three, and the diﬀerences among these models are particularly transparent in the Lagrangean description. The Chu formulation diﬀers from the statistical and Lorentz formulation only in the choice of magnetization. This resulted in the selection of diﬀerent auxiliary ﬁelds, which are related to the basic ﬁelds via the Maxwell-Lorentz aether relations (5.2.9) and (5.3.12). As a result, Hα and Eα are the two basic electromagnetic ﬁeld vectors in the Chu formulation, while Bα and Eα are those of the statistical or Lorentz formulation. In

134

5 Material Description

this connection it is interesting to note that the Lagrangean electric ﬁeld Eα is the same in all formulations. This was not so in the spatial description where EiC and EiL = EiS are diﬀerent variables. The same is also true for the Lagrangean magnetic ﬁeld strength Hα ; it is the same variable in both the Chu and the Minkowski formulation. Another noteworthy point is that the Maxwell equations are formally the same as those in the spatial version of the statistical or Minkowski formulation. Because the transformations (5.6.3) and (5.6.4) relating the systems (5.6.1) and (5.6.2) to (5.6.6) are so simple, it is now also plausible why the Maxwell-Minkowski formulation served as connecting piece between the two-dipole and the statistical and the Lorentz formulations. In performing actual calculations in electromagnetism it is advantageous to introduce electromagnetic potentials by satisfying those Maxwell equations identically that do not involve the charge Q and the current densities Jα . Mathematically, this is achieved by introducing two potentials Aα and Φ, such that (5.6.7) Bα = eαβγ Aγ,β , and Eα = Φ,α − A˙ α . Since Bα is a basic ﬁeld in all but the Chu formulation, the replacement of the Maxwell equations by the corresponding equations for the potentials Aα and Φ and for the charge Q is easier in the Lorentz, the statistical and the Maxwell–Minkowski formulation than it is in the Chu formulation. This may be regarded as a disadvantage of the Chu formulation. The above discussion is only concerned with the Maxwell equations and leaves all mechanical balance laws aside. Yet, the basic advantages of the diﬀerent formulations are drawn from the expressions of electromagnetic body force, body couple and energy supply. These are listed in (5.2.11) and (5.2.12) (two-dipole models), (5.3.16) (Lorentz formulation), (5.3.19) (statistical formulation) and (5.3.23) (Maxwell-Minkowski formulation). Of all these formulations the two-dipole model with symmetric stress tensor and the Lorentz formulation led to electromagnetic body force and energy supply expressions, which are particularly simple and easy to interpret (see the corresponding expressions (5.2.12) and (5.3.16)). This is not so for all other formulations, although it is known that their interaction terms have a clear physical meaning and are based on a clear method of derivation. It is also interesting to note that the two most simple formulations are those with no electromagnetic body couple and with symmetric Cauchy stress. Hence, since all formulations are non-relativistically equivalent anyhow, future calculations should be performed with either one of these simplest formulations. The argument on electromagnetic potentials given above favors the Lorentz formulation. We shall come back to this point in Chap. 6. As was done for the Maxwell equations, we also wish to recapitulate the mechanical balance equations and to comment on the equivalence properties in the ﬁve presented models. Starting with the Chu models, we showed in

5.6 Recapitulation and Comparison

135

Sect. 5.2 already that these two models are equivalent and, therefore, we only list the balance equations for model II: ¨i = ρ0 x

II

−1 Tiα,α + Fαi {(Q − Pβ,β )Eα + eαβγ (Jβ + P˙ β )Bγa

C −µ0 Mβ,β Hα } + ρ0 Fiext . II

(5.6.8)

T[iα Fj]α = 0 ,

ρ0 Θη˙ = Jα Eα − Qα,α + ρ0 rext . Here, the ﬁrst and the second equation follow by substituting (5.2.12)1,2 into (2.2.32)1 and (2.2.32)2 . The corresponding mechanical balance equations for model I are most easily obtained if in (5.6.8) the stress tensor II Tiα is replaced by I Tiα ; this is achieved through use of (5.2.22). Next we pass on to model V, or the Lorentz model. Its mechanical balance laws emerge if (5.3.16) and (5.3.30) are substituted into (2.2.32). When this is done care must be observed that the heat ﬂux vector Qα in (5.3.30) is correctly selected; it is QSα of the statistical model, rather than QL α as originally introduced in the Lorentz model . We thus obtained ¨i = ρ0 x

V

−1 Tiα,α + Fαi {(Q − Pβ,β )Eα + eαβγ (Jβ + P˙ β )Bγ L ext +(ML , α,β − Mβ,α )Bβ } + ρ0 Fi

V

(5.6.9)

T[iα Fj]α = 0 ,

ρ0 Θη˙ = Jα Eα − Qα,α + ρ0 rext . The next model is model IV, but again there is no need to list the results for this model explicitly, because the only diﬀerence between models IV and V lies in the stress tensors and consequently the balance equations for model IV may be obtained by merely introducing into (5.6.9) IV Tiα as given by (5.3.26). Needless to state that the two models have been shown to be equivalent already in Sect. 5.3. There remains the presentation of model III. Because of its complexity as compared to other models it is presented last. Its mechanical balance equations emerge when (5.4.5) is substituted into (2.2.32). This yields ¨i = ρ0 x

III

−1 Tiα,α + Fαi {Q Eα + eαβγ Jβ Bγ + Pβ Eβ,α

˙ ˙ +µ0 MC β Hβ,α + eαβγ (Dβ Bγ + Dβ Bγ )} −1 ext +Fβi,j Fjα (Pα Eβ + µ0 MC , α Hβ ) + ρ0 Fi III

−1 T[iα Fj]α = F[iα Fβj] (Pα Eβ + µ0 MC α Hβ ) ,

ρ0 Θη˙ = Jα Eα − Qα,α + ρ0 rext .

(5.6.10)

136

5 Material Description

As already stated in Sect. 5.4, model III is identical to model I, and, hence, the only diﬀerence in the formulations of models III and II stems from the diﬀerence in the deﬁnition of the stress tensors, which are related to each other according to (see (5.4.13) or (5.2.22)) III

Tiα =

II

−1 Tiα − Fβi (Pα Eβ + µ0 MC α Hβ ) .

(5.6.11)

From the above considerations we conclude that the only necessary step for a completion of the comparison of the ﬁve models is a comparison of models II and V. To perform this comparison, notice that in view of (5.6.8) and (5.6.9) the balance laws of moment of momentum and energy are identical in the two formulations. Consequently, at most the balance laws of linear momentum can diﬀer; and the diﬀerence can at most be a diﬀerence in stress. This diﬀerence must, furthermore, be such that the balance of moment of momentum is met; in other words the diﬀerence between the second Piola-Kirchhoff tensors II P P Tαβ and V Tαβ must be a symmetric tensor. In order to ﬁnd this diﬀerence the balance equations (5.6.8)1 and (5.6.9)2 must be used. However, this step becomes much easier if for the electromagnetic body force the respective representations in terms of the Maxwell stress tensor are used; see (5.2.17)1 and (5.3.19)1 . We then obtain V

Tiα −

II

Tiα =

II

M Tiα −

V

M Tiα

−1 −1 = Fβi (Bαa Hβ − Bα Hβa ) − 12 Fαi (Bβa Hβ − Bβ Hβa ) .

(5.6.12)

To eliminate from this expression the auxiliary ﬁelds use must be made of (5.2.9), (5.3.12) and (5.4.2); we may then derive the expressions −1 Hγ Hβ , Bαa Hβ = µ0 |J|Cαγ

Bα Hβa =

1 −1 −1 |J |Cβγ Bγ Bα = µ0 |J|Cαγ Hγ Hβ + µ0 MαC Hβ µ0 −1 +µ0 Cαγ Cβδ Hγ MδC + µ0 |J −1 |Cβγ MαC MγC .

(5.6.13)

Substitution into (5.6.12) yields V

Tiα =

II

−1 −1 Tiα − µ0 Fβi MαC Hβ − µ0 Cαγ Fiβ Hγ MβC

−1 −µ0 |J −1 |Fiβ MαC MβC + µ0 Fαi MβC Hβ

(5.6.14)

−1 + 12 µ0 |J −1 |Fαi Cβγ MβC MγC , −1 and after a multiplication with Fβi V

P Tαβ =

II

−1 −1 P −1 Tαβ − µ0 Cβγ MαC Hγ − µ0 Cαγ Mβ Hγ + µ0 Cαβ Mγ Hγ

−1 −µ0 |J −1 |MαC MβC + 12 µ0 |J −1 |Cαβ Cγδ MγC MδC ,

(5.6.15)

5.6 Recapitulation and Comparison

137

P P from which it is easily seen that the diﬀerence between V Tαβ and II Tαβ forms indeed a symmetric tensor. Needless to say that this relation corresponds to an equation that was established in Chap. 4 and relates the Cauchy stresses V tij and II tij (see Table 4.1). P P is expressed in terms of II Tαβ In (5.6.15), the Lorentzian stress V Tαβ II P V P and the Chu-variables. The reverse relation expresses Tαβ in terms of Tαβ and the Lorentz-variables. This latter expression is obtained from (5.3.10), (5.3.14)2 and (5.3.12)2 and reads II

P Tαβ =

V

−1 L −1 L P −1 L Tαβ + Cαγ Mγ Bβ + Cβγ Mγ Bα − Cαβ Mγ Bγ

−1 −1 L L −1 −1 L L −µ0 |J|Cαγ Cβδ Mγ Mδ + 12 µ0 |J|Cαβ Cγδ Mγ Mδ .

(5.6.16)

The above equations and similar relations of the previous models are necessary conditions, which must be satisﬁed in order that the models be equivalent. If these relations hold equivalence goes as far as the local balance laws of linear and angular momentum, energy and the corresponding jump conditions are concerned. The conditions are not suﬃcient, however, because the constitutive relations and therefore also the thermodynamic requirements impose further conditions. In the following these thermodynamic requirements will be discussed. It follows from the foregoing considerations that in the constitutive theories of models I, II and III the internal energies must be identical, i.e. I

(ρ0 U ) =

II

(ρ0 U ) =

III

(ρ0 U ) ,

(5.6.17)

if the models are to be equivalent. The same holds for models IV and V: IV

(ρ0 U ) = V(ρ0 U ) .

(5.6.18)

However, and as could already be expected from the corresponding results of Chap. 4, relation (5.6.15) can only be satisﬁed simultaneously with the constitutive equations (5.2.26) and (5.3.29) provided that the internal energies for models II and V diﬀer. To derive the corresponding relation note that according to Chap. 4 V

(ρU ) =

(ρU ) − µ0 Hi Mi − 12 µ0 Mi Mi ,

II

which in Lagrangean notation reads V

(ρ0 U ) =

(ρ0 U ) − µ0 Hα MαC − 12 µ0 |J −1 |Cαβ MαC MβC . (5.6.19)

II

If for models II and V we use constitutive relations of the form (5.2.53) and (5.3.35) and if we introduce the Helmholtz free energies (5.2.54) and (5.3.36), then (5.6.19) is equivalent to ˇ = (ρ0 ψ)

V

¯ − 1 µ0 |J −1 |Cαβ MC MC . (ρ0 ψ) α β 2

II

(5.6.20)

138

5 Material Description

This equation may be used in (5.2.55) and (5.3.37) to obtain expressions for the entropy, polarization, magnetization and the stress in the respective formulations. These must then satisfy the identities derived above, and if they do, the models are equivalent. The proof on the basis of (5.6.20), (5.2.55) and (5.3.37) is straightforward and thus we leave it to the reader. However, we shall come back to the consequences of this statement when deﬁning material coeﬃcients in the various formulations. Finally, we would like to point out that a major goal of the derivation of the material description was in the presentation of a correct approach in linearizing ﬁeld equations. The Lagrangean formulation is useful from just this practical point of view. Indeed, once the equations are known in their material description the transformations ∂ ∂ → ∂xi ∂Xα

and

d ∂ → ∂t dt

need no longer be performed, because all variables are per se already functions of Xα and t. This means that any perturbation approximation is much easier to be carried out when the equations are written in material rather than spatial coordinates. All linearization procedures performed so far were in the spatial description (see e.g. Toupin [241], Hutter and Pao [91] and Van de Ven [249]). The treatments of Toupin and Hutter and Pao are approximate, however, insofar as they contain ad hoc assumptions that cannot be justiﬁed on the basis of non-relativistic arguments. This was pointed out by Van de Ven, who also presents the correct solution. All the difﬁculties Toupin and Hutter and Pao were faced with disappear in the material description and no ad hoc assumptions must be introduced here. Corroboration of this will be given in the next chapter.

5.7 Approach to a Uniﬁed Constitutive Theory In the preceding section we demonstrated that all theories of deformable bodies in the electromagnetic ﬁelds which have the complexity of thermoelastic polarizable and magnetizable solids are non-relativistically equivalent, provided that the respective constitutive functions for the internal energy or the free energy satisfy certain relationships. Similar equivalence statements were already established in the Eulerian description so that a new proof was not a necessity except, perhaps, that it led to equivalence conditions expressed in the Lagrangean variables; these are very useful relationships. To recapitulate them brieﬂy, recall that in order to enforce equivalence of formulations I, II and III the free energies must in all these formulations be the same functions of the same variables. A similar statement also holds for models IV and V, but to achieve equivalence between the groups (I, II, III) and (IV,V) the corresponding free energies must be related by (5.6.20), viz.

5.7 Approach to a Uniﬁed Constitutive Theory

ρ0 ψˇ = ρ0 ψ¯ − 12 µ0 |J −1 |Cαβ MαC MβC .

139

(5.7.1)

Here, ψ¯ is the free energy function in any one of the formulations I, II or III, and, correspondingly ψˇ is that of the models IV and V; they are deﬁned in (5.2.54) and (5.3.36), respectively. Accordingly, and apart from a dependence on Cαβ and Θ, ψˇ is a function of (Eα , Bα ), whereas ψ¯ depends on (Eα , Hα ) as does MC α because (see (5.2.55)3 ) MC α =−

ρ0 ∂ ψ¯ . µ0 ∂Hα

Equation (5.7.1) can therefore also be written as ˇ αβ , Eα , Bα , Θ) = ψ(C ¯ αβ , Eα , Hα , Θ) ψ(C ¯ ∂ ψ ∂ ψ¯ 1 ρ0 −1 |J |Cαβ (Cαβ , Eα , Bα , Θ) . − 2 µ0 ∂Hα ∂Hβ

(5.7.2)

In this identity, the left-hand side is a function of Bα , in contrast to the expression on the right-hand side, which is a function of Hα . In view of (5.4.2)2 and (5.2.55)3 we may, however, express Bα as Bα = −ρ0 so that

∂ ψ¯ −1 + µ0 |J|Cαβ Hβ , ∂Hα

(5.7.3)

ˇ αβ , Eα , µ0 |J|C −1 Hβ − ρ0 ∂ ψ/∂H ¯ ψ(C α , Θ) = αβ

(5.7.4) ¯ ¯ ¯ αβ , Eα , Hα , Θ) − 1 ρ0 |J −1 |Cαβ ∂ ψ ∂ ψ , ψ(C 2 µ0 ∂Hα ∂Hβ ¯ ¯ in which the arguments in ∂ ψ/∂H α are, of course, the same as in ψ. For ˇ equation (5.7.4) must be satisﬁed identically if the given functions ψ¯ and ψ, formulations (I, II, III) and (IV,V) are to be equivalent. If, on the other hand, only one of the functions ψˇ or ψ¯ is given, then (5.7.4) is a functional diﬀerential equation to determine the other. The solution to this will, in general, ¯ say. We be very complex. We shall not try to solve it for a given function ψ, ˇ A shall rather exploit (5.7.4) as common forms of the free energies ψ¯ and ψ. very popular procedure is to write these functions as polynomial expressions of the independent variables. These polynomials must be regarded as truncated Taylor series expansions about a state of zero electromagnetic ﬁelds and zero deformation of a general functional relationship for the free energy. Now, the equivalence of the various models was guaranteed in the above for the general theory using a representation for the constitutive equations of the free energy not restricted by any means. Otherwise stated, if one attempts to establish equivalence statements between two given theories for which the free energies are polynomials truncated at the quadratic terms, the two theories might very well be non-equivalent simply because equivalence would in

140

5 Material Description

one formulation require the inclusion of cubic, quartic or even higher order terms. The point just raised is important, because it illustrates that, strictly, fully equivalent theories might become non-equivalent, because in both formulations one insists in too restrictive energy expressions. Nonetheless, we can keep the usual polynomial representations and still claim equivalence, but if we do so this is only in the following restricted sense: We formally interpret the polynomials as truncated Taylor series expansions. With the use of the latter equivalence of two formulations can be established exactly; it amounts to a comparison of the polynomial coeﬃcients. If in these Taylor series expansions we restrict ourselves to terms of a certain order, then equivalence can be established approximately. To illustrate the above point more clearly, consider the following somewhat academic example: Let f (x, y) =

∞ )

aµν xµ y ν ,

g(x, z) =

µ,ν=0

∞ )

bµν xµ z ν ,

µ,ν=0

and assume that for some reason z = x + y and f = g. Then by the binomial theorem we may set ν ) ν k ν−k ν ν z = (x + y) = , x y k k=0

and therefore ∞ )

µ ν

aµν x y =

µ,ν=0

∞ ) µ,ν=0

bµν

ν ) ν k=0

k

xk+µ y ν−k .

The latter equation can also be written as " µ # ∞ ) ) ν + k b(µ−k)(ν+k) − aµν xµ y ν = 0 , k µ,ν=0 k=0

so that the coeﬃcient functions must satisfy the relations µ ) ν+k aµν = b(µ−k)(ν+k) . k k=0

These identities must hold for all positive integers µ and ν. If we truncate the above polynomial representations at ν = N and µ = M they can still be satisﬁed, but then fM N and gM N , which denote the truncated expressions fM N (x, y) =

N M ) ) µ=0 ν=0

aµν xµ z ν ,

gM N (x, y) =

N M ) ) µ=0 ν=0

bµν xµ z ν ,

5.7 Approach to a Uniﬁed Constitutive Theory

141

are no longer exactly equal, but only in the sense that fM N = gM N + O(xM +1 ) + O(y N +1 ) , holds. For instance, if f12 (x, y) = a00 + a01 y + a02 y 2 + a10 x + a11 xy + a12 xy 2 , g12 (x, z) = b00 + b01 z + b02 z 2 + b10 x + b11 xz + b12 xz 2 , and g12 is expressed in terms of x and y, then g12 should contain quadratic and cubic terms in x, whence follows that f12 and g12 cannot be identical except in the above mentioned approximate sense. In fact, one obtains g12 = f12 + (b11 + b02 + 2b12 y)x2 + b12 x3 . The above considerations may look somewhat artiﬁcial to the novel reader, yet they are important, and we would like to illustrate them using the most simple example that accounts for magnetoelastic interactions. For that purpose we restrict ourselves to conditions of isotropy and to free energies, which are at most of quartic order in the electric and magnetic ﬁeld quantities and of quadratic order in the temperature diﬀerence θ = Θ − Θ0 . Here, Θ0 denotes a reference temperature. Moreover, we assume small deformations, so that it suﬃces to write the free energy as a quadratic function of the Lagrangean deformation tensor Eαβ = 12 (Cαβ − δαβ ) .

(5.7.5)

With these limitations we may choose the following representations for the free energy functions: for model II 1 1 1 ψ¯ = ¯(m) Hα Hα + ¯(m) (Hα Hα )2 + ¯(e) Eα Eα 2χ 4χ 2χ 2ρ0 4ρ0 2ρ0 1 1 ¯ (m) 1 ¯ (e) L Hα Hα θ + L Eα Eα θ ¯(e) (Eα Eα )2 − 12 ¯cθ2 + 4χ 4ρ0 2ρ0 2ρ0 1 ¯(m) 1 ¯(e) + Hα Hβ + Eα Eβ − ν¯δγδ θ Eγδ b b 2ρ0 αβγδ 2ρ0 αβγδ

+

+

1 ¯cαβγδ Eαβ Eγδ , 2ρ0

(5.7.6)

142

5 Material Description

for model V 1 1 1 ˇ(m) Bα Bα + ˇ(m) (Bα Bα )2 + ˇ(e) Eα Eα ψˇ = 2χ 4χ 2χ 2ρ0 4ρ0 2ρ0 1 1 ˇ (m) 1 ˇ (e) L Bα Bα θ + L Eα Eα θ ˇ(e) (Eα Eα )2 − 12 cˇθ2 + 4χ 4ρ0 2ρ0 2ρ0 1 ˇ(m) 1 ˇ(e) Bα Bβ + Eα Eβ − νˇδγδ θ Eγδ + b b 2ρ0 αβγδ 2ρ0 αβγδ

+

+

1 cˇαβγδ Eαβ Eγδ . 2ρ0

(5.7.7)

Before we proceed it is worthwhile to look at these expressions more closely. The polynomial representations for the free energy start with quadratic terms. A constant term is left out, because it is immaterial, and linear terms are discarded, because at zero deformation, zero temperature diﬀerence and zero electromagnetic ﬁelds no stress and no polarization and magnetization should be present. In other words we assume the body to possess a natural unstrained state. Cubic terms are also present in (5.7.6) and (5.7.7), but in this regard the polynomials are not complete, since products such as Eαβ Eγδ Hε , Eαβ , Eγδ Eε , and Eαβ Eγδ θ are missing. We believe that these omissions are justiﬁed, because usually the ﬁeld-free elasticities are much more important than their change by the electromagnetic ﬁelds. We have further neglected electromagnetic coupling terms. In the ensuing analysis our aim is to investigate in what sense the above representations would allow an identical satisfaction of (5.7.1). In view of the above general remarks we expect that a full equivalence cannot be established, so diﬀerences will show up in the higher-order terms of (5.7.6) and (5.7.7). To ﬁnd the equivalence conditions we substitute (5.7.6) into the ﬁrst term on the right-hand side of (5.7.1) and rewrite the latter by expressing the second ¯ To this end, note that term as a function of the independent variables of ψ. |J −1 |Cαβ = δαβ (1 − Eγγ ) + 2Eαβ + O(E 2 )δαβ + nαβγδ Eγδ + O(E 2 ) , (5.7.8) where nαβγδ = −δαβ δγδ + δαγ δβδ + δαδ δβγ .

(5.7.9)

In view of (5.2.55)3 and (5.7.6) we may also write µ0 MC α = −ρ0

∂ ψ¯ ∂Hα

¯(m) Hβ Eγδ . ¯ (m) Hα θ − b = − 2χ ¯(m) Hα − 4 χ ¯(m) Hβ Hβ Hα − L αβγδ (5.7.10) Using these relations in (5.7.1) we ﬁnd that the right-hand side of the latter may be written as

5.7 Approach to a Uniﬁed Constitutive Theory

143

¯(m) 1 2χ ¯(m) Hα Hα ψˇ = 1− 2χ 2ρ0 µ0 ¯(m) 1 2χ (m) + ¯ 1−4 (Hα Hα )2 4χ 4ρ0 µ0 1 1 + ¯(e) Eα Eα + ¯(e) (Eα Eα)2 − 12 ¯cθ2 2χ 4χ 2ρ0 2ρ0 ¯(m) ¯ (m) 1 1 ¯ (e) 2χ L Hα Hα θ + L Eα Eα θ + 1−2 2ρ0 µ0 2ρ0 ! (5.7.11) 2 ¯(m) ¯(m) ¯(m) 1 2χ 2χ 1−2 nαβγδ Hα Hβ Eγδ + bαβγδ − 2ρ0 µ0 µ0 1 ¯(e) 1 + bαβγδ Eα Eβ Eγδ − ν¯θEγγ + ¯cαβγδ Eαβ Eγδ 2ρ0 2ρ ¯ (m)2 ¯ (m) L L ¯(m) Hα Hβ Eγδ θ − 2χ − Hα Hα θ2 + ¯(m) nαβγδ − b αβγδ 2ρ0 µ0 ρ0 µ0 2 ¯ (m) L nαβγδ Hα Hβ Eγδ θ2 + higher order terms . − 2ρ0 µ0 Here and henceforth all terms of order Hε Eαβ Eγδ ,

Eε Eαβ Eγδ ,

θEαβ Eγδ ,

and higher are neglected. This is consistent with our basic assumption that third-order terms involving the square of the deformation tensor are discarded. In contrast to (5.7.6) or (5.7.7) the above representation for ψˇ contains also mixed fourth-order terms (the last three expressions on the righthand side). For reasons described above, these should be neglected as well. The occurrence of these higher-order terms is corroboration for our earlier statement that full equivalence might simply be impossible, because the free energies in the respective formulations are too restrictive. However, it is apparent from the above calculation that a full equivalence could be achieved if the entire Taylor series expansion would be kept. The functional relationship for ψˇ is still not in the appropriate form for comparison with (5.7.7). For that purpose Bα in (5.7.7) must be replaced by Hα . This replacement is accomplished by using (5.4.2) and (5.7.10), and it yields ¯ (m) Hα θ ¯(m) )Hα − 4 χ ¯(m) Hβ Hβ Hα − L Bα = µ0 (1 − 2 χ (m)

−(bαβγδ + µ0 nαβγδ )Hβ Eγδ ¯ (m) Hα θ = µHα − 4 χ ¯(m) Hβ Hβ Hα − L (m)

¯ −(b αβγδ + µ0 nαβγδ )Hβ Eγδ , where we have set

(5.7.12)

144

5 Material Description

µ0 − 2 χ ¯(m) = µ ,

or

¯(m) 2χ

= µ0 − µ .

(5.7.13)

If (5.7.12) is substituted into (5.7.7) and the resulting expression is rearranged (thereby neglecting third and fourth order terms as was done above) we obtain 1 2 (m) ψˇ = µ 2χ ˇ Hα Hα 2ρ0 1 (m) (m) µ4 4 χ (Hα Hα )2 + ˇ(m) − 4µ 2 χ ˇ4 χ ˇ 4ρ0 1 1 + ˇ(e) Eα Eα + ˇ(e) (Eα Eα )2 − 12 ˇcθ2 2χ 4χ 2ρ0 4ρ0 1 1 ˇ (e) ˇ (m) − 2µL ¯ (m) 2 χ L Eα Eα θ + (µ2 L ˇ(m) )Hα Hα θ + 2ρ0 2ρ0 1 (m) ˇ(m) − 2µ b ¯(m) + µ0 nαβγδ 2 χ µ2 b Hα Hβ Eγδ ˇ + αβγδ αβγδ 2ρ0 1 ˇ(e) 1 + Eα Eβ Eγδ − νˇθEγγ + ˇcαβγδ Eαβ Eγδ , b 2ρ0 αβγδ 2ρ0

(5.7.14)

which is expressed in terms of the coeﬃcients (ˇ) as well as (¯). Identifying (5.7.11) with (5.7.14), we obtain ˇ(m) = 2 χ ¯(m) = µ0 − µ =: −µ0 2 χ(m) , µ0 µ 2 χ ˇ(e) 2χ

= 2χ ¯(e) =: − 2 χ(e) ,

1 ¯(m) , 4 χ ¯(e) = 4 χ ˇ(e) =: − 4 χ(e) , 4χ µ4 cW ˇ (m) = L ¯ (m) , L ˇ (e) = L ¯ (e) =: L(e) , , µ2 L ˇc = ¯c =: Θ0 (m) ˇ(m) = 1 b ¯(m) − µ0 2 χ (2 + 2 χ(m) )nαβγδ , b αβγδ µ2 αβγδ µ2 (e) (e) (e) ¯ ˇ =b =b , b ˇ(m) 4χ

=

αβγδ

αβγδ

νˇ = ν¯ = ν,

(5.7.15)

αβγδ

ˇcαβγδ = ¯cαβγδ = cαβγδ .

Several of the coeﬃcients occurring in the above equations can, without confusion, be given speciﬁc names, as for instance 2χ

(m)

magnetic susceptibility ,

µ

magnetic permeability ,

2χ

(e)

cW (e)

L

, 4χ

(e)

second- and fourth-order electric susceptibility , speciﬁc heat , thermoelectric constant ,

5.7 Approach to a Uniﬁed Constitutive Theory

145

(e)

bαβγδ electrostrictive constants , ν

thermoelastic constant ,

cαβγδ elastic constants . ¯ (m) More diﬃculties arise, however, for L(m) , because it is not clear whether L (m) ˇ or L should be called thermomagnetic constant. A similar statement also ¯(m) and 4 χ ˇ(m) . Since the diﬀerence holds for the anisotropy coeﬃcients 4 χ (m) (m) ¯ ˇ between b αβγδ and bαβγδ is essential, the situation is even more drastic for (m)

bαβγδ , the coeﬃcients usually attributed with the notion of magnetostric¯(m) a magnetostrictive tion. In the Chu formulation one is inclined to call b αβγδ ˇ(m) instead. constant; in the Lorentz formulation this is the case for b αβγδ For isotropic materials the relations (5.7.15)8 can still somewhat be simpliﬁed, since for this special group of materials (m)

(m)

(m)

bαβγδ = b1 δαβ δγδ + b2 (δαγ δβδ + δαδ δβγ ) must hold. Substituting this into (5.7.15)8 we obtain ˇ(m) = 1 b ¯(m) + µ0 2 χ(m) ( 2 χ(m) + 2) , b 1 µ2 1 ˇ(m) = 1 b ¯(m) − µ0 2 χ(m) ( 2 χ(m) + 2) . b 2 1 2 µ

(5.7.16)

(5.7.17)

From these expressions it is now evident that a unique deﬁnition of magnetostrictive constants is not possible by merely establishing the respective polynomial representation of the free energy function. What would be needed, is a simple experiment in which magnetostrictive eﬀects could uniquely be deﬁned and in which the magnetostrictive constants could be measured, which then must be independent of the model chosen. At this point we must warn the reader to take the above conclusions as the ultimate truth. The results are special insofar as they hold for isotropic bodies and no electromagnetic coupling. For instance that all electric coeﬃcients in the energy expressions are identical is a consequence of the omission of electromagnetic coupling terms (these are terms such as χ ¯(em) Eα Hα and (em) ¯ bαβγδ Eα Hβ Eγδ ). Any generalization to these more complicated polynomial expressions is straightforward, however, and will be left to the reader. There still remains the evaluation and comparison of the constitutive relations (5.2.55) and (5.3.37). To begin with, let us look more closely at the entropy. According to (5.2.55)1 and (5.7.6), we obtain η=−

II

θ ∂ ψ¯ 1 ¯ (m) L(e) L Hα Hα − = cW − Eα Eα + νEγγ . (5.7.18) ∂θ Θ0 2ρ0 2ρ0

On the other hand, straightforward evaluation of (5.3.37)1 on the basis of (5.7.7) gives

146

5 Material Description

η=−

V

θ ∂ ψˇ 1 ˇ (m) L(e) L Bα Bα − = cW − Eα Eα + νEγγ , (5.7.19) ∂θ Θ0 2ρ0 2ρ0

which with the aid of (5.7.12) and (5.7.15) becomes

θ 1 ¯ (m) 2 ¯ (m) 2 (m) V L η = cW − ¯ Hγ Hγ − L θ δαβ 1 − 4 χ Θ0 2ρ0 µ µ 2 ¯(m) L(e) − (bαβγδ + µ0 nαβγδ )Eγδ Hα Hβ − Eα Eα + νEγγ µ 2ρ0 + higher-order terms .

(5.7.20)

Mere comparison of (5.7.18) and (5.7.20) shows that II η diﬀers from V η. Hence, use of (5.7.15) has not led to identical expressions for the entropy. The diﬀerence arises, because of the terms (Hα Hα )2 , Hα Hα θ, Hα Hβ Eγδ , and still higher-order terms, which in the energy expression can be traced back to the fourth-order terms (Hα Hα )2 θ, Hα Hα θ2 , and Hα Hβ Eαβ θ. In the process of the transformation of the energy expressions these fourth-order terms were omitted (and they must be, if the polynomials are interpreted as truncated Taylor series expansions). Similar discrepancies also occur when the other constitutive quantities derivable from the free energy are determined. These are given by Pα =

II

Pα =

V

Pα = 2 χ(e) Eα + 4 χ(e) Eβ Eβ Eα − L(e) Eα θ (5.7.21)

(e)

−bαβγδ Eβ Eγδ , (m) ¯ (m) Hα θ Hα − 4 χ ¯(m) Hβ Hβ Hα − L µ0 MC α = µ0 2 χ (m)

¯ −b αβγδ Hβ Eγδ , ML α =

2χ

(m)

µ

(5.7.22)

ˇ(m) Bβ Eγδ , (5.7.23) ˇ (m) Bα θ − b Bα − 4 χ ˇ(m) Bβ Bβ Bα − L αβγδ

(m)

(e)

II

¯ TP αβ = bγδαβ Hγ Hδ + bγδαβ Eγ Eδ − νδαβ θ + cαβγδ Eγδ ,

(5.7.24)

V

(e) ˇ(m) TP αβ = bγδαβ Bγ Bδ + bγδαβ Eγ Eδ − νδαβ θ + cαβγδ Eγδ .

(5.7.25)

They can be transformed into each other by neglecting all inconsistent terms as was done above for the entropy. This completes our transformation of the theories of group (I, II, III) into those of group (IV,V). The calculations show that full equivalence may be destroyed by a too special choice of the energy functions. However, the calculations have simultaneously demonstrated how the material coeﬃcients of one theory can be related to those of another. These questions are of immense practical importance and will be reconsidered in Chap. 6.

6 Linearization

6.1 Statement of the Problem The governing dynamical equations of ﬁeld-matter interaction in thermoelastic materials as outlined in the previous chapters are highly nonlinear. Generally, it is hardly possible to ﬁnd exact solutions even if the most simple problems that are still of some physical relevance are attacked. As stated in Chap. 5 already, one of the major disadvantages, namely that the equations are given in the spatial description, while boundary conditions for a solid body are usually prescribed in the reference conﬁguration, has been removed by the introduction of a consistent material description for both the ﬁeld equations and the jump and boundary conditions. Nevertheless, the resulting equations are still highly nonlinear, and this implies that some approximation scheme must be found. To render the equations amenable to direct analysis, they will be linearized with respect to some intermediate state. We suppose that in the intermediate state the position of a material point, initially at X, is given by ξ (ξα , α = 1, 2, 3). The total motion of the particle from its initial position X to its ﬁnal position x is then decomposed into the motion from the reference state to ¯, the intermediate state, characterized by the displacement vector U ¯ =ξ−X , U

(6.1.1)

and the motion from the intermediate state to the present state with displacement u u=x−ξ , (6.1.2) (i.e. from now on u is not the total displacement, but the displacement from ξ to x only). In the above, ξ was not speciﬁed. In principle, any continuous map X → ξ can be considered suitable for this intermediate conﬁguration of the body. However, it will be assumed that the problem at hand suggests a natural deﬁnition of this state. For the time being we assume that it is known or at least determinable. We further assume that the position of a material point in the intermediate state ξ is close to its ﬁnal position x. Based on this closeness the motion and, more generally, all ﬁelds may be decomposed into two parts. One of these parts represents the ﬁelds when u = 0, the other one is due to the K. Hutter et al.: Electromagnetic Field Matter Interaction in Thermoelastic Solids and Viscous Fluids, Lect. Notes Phys. 710, 147–197 (2006) c Springer-Verlag Berlin Heidelberg 2006 DOI 10.1007/3-540-37240-7 6

148

6 Linearization

perturbations from the ξ-state to the x-state. Because these perturbations are assumed to be small, the governing equations can be linearized in these perturbations. This linearization procedure can be performed in a consistent way. For instance, if a body is initially subjected to large biasing electromagnetic ﬁelds and wave propagation or vibration properties of this body are investigated, then our linearization procedure will be applicable (e.g. [233, 248] and [93, 94]). Of the same nature are magnetoelastic stability problems [152, 250]. To describe the linearization procedure into more detail, all ﬁeld variables will be decomposed into two parts. Those in the intermediate state are labelled with an overhead bar, whereas the perturbations on this state are indicated by lower case letters, e.g. ¯ α + bα , Bα = B where

b = O(ε) , B

¯ α + eα , Eα = E e = O(ε) , E

etc. ,

(6.1.3)

(6.1.4)

and where ε denotes a small positive quantity (0 < ε 1). The norms in (6.1.4) may conveniently be deﬁned as a2 := limsup(a2 (τ )) , 0≤τ ≤t

(6.1.5)

where τ = 0 is the time at which the process started and t is the current time. Moreover, the displacements from ξ to x and their material time derivatives are assumed to be small in the sense that && && && ∂u && ˙ ||u|| && && and = O(ε) (6.1.6) && ∂ξ && = O(ε) v0 where v0 is some characteristic wave speed. In the Lagrangean description any diﬀerentiation with respect to X or t falls directly onto the respective variables as decomposed in (6.1.3). For instance ¯ α,α + bα,α . Bα,α = B This is particularly easy and convenient, and to see this let us ﬁrst brieﬂy investigate the Eulerian formulation. In this case all equations must be traced back to the intermediate conﬁguration. In particular, all derivatives with respect to the present coordinates must be expressed in terms of the intermediate coordinates by means of the transformation rules, which we shall now brieﬂy outline, (although lateron they will not be used). To this end, let f be any physical quantity. It may be regarded as a function of the variables (X, t), (ξ, t) or (x, t); thus f = fˆ(X, t) = f˜(ξ, t) = fˇ(x, t) .

(6.1.7)

6.1 Statement of the Problem

149

Depending on which representation we choose, we thus have ∂ fˇ ∂ f˜ ∂ ξˇα ∂u ˇα ∂ f˜ = = δiα − . ∂xi ∂ξα ∂xi ∂xi ∂ξα In particular with f ≡ uα we obtain ˜α ∼ ∂u ˇα ∂u ˜α ∂u ˇβ ∂ u = δiβ − , = δij ∂xi ∂xi ∂ξβ ∂ξβ whence follows

∂ fˇ ∼ ∂u ˜α ∂ f˜ . = δiβ δαβ − ∂xi ∂ξβ ∂ξα

(6.1.8)

Similarly, for the material time derivative we obtain df ˆi ∂ fˆ ∂ f˜ ∂ f˜ ∂ ξˆα ∂ fˇ ∂ fˇ ∂ x = = + = + , dt ∂t ∂t ∂ξα ∂t ∂t ∂xi ∂t from which one easily deduces that ∂ fˇ ∂ f˜ ∂ f˜ ∂ ξˆα ∂ f˜ ∂ ξˇα ∂ x ∂ f˜ ∂ f˜ ˆi = + − = − ∂t ∂t ∂ξα ∂t ∂ξα ∂xi ∂t ∂t ∂ξα

∂ ξˆα ˆi ∂ ξˇα ∂ x − ∂xi ∂t ∂t

.

Here, in each of the occurring functions we have indicated the functional dependencies. With the obvious deﬁnitions ∂ ξˆα := ξ˙α ∂t

and

∂x ˆi := x˙ i = u˙ i + ξ˙α δiα ∂t

this now becomes ˜ ∂f ∂ f˜ ∂ fˇ ∂u ˜α ˙ ˙ u˙ i + δiγ ξγ − ξα = − δiβ δαβ − ∂t ∂t ∂ξβ ∂ξα ˜ ˜ ∂ f ∂ u ˜ ∂ f α ∼ − u˙ α − ξ˙β . = ∂t ∂ξβ ∂ξα

(6.1.9)

The above formulas (6.1.8) and (6.1.9) must be applied in all ﬁeld equations whenever space and time derivatives of physical quantities occur. That this is very tedious can be seen from the fact that for a dynamical theory it has been tried by several authors in the past as e.g. Toupin [241], Hutter and Pao [91] and Van de Ven [249]. However, except for the procedure of Van de Ven none of these is completely correct. Toupin,Toupin63, for instance, seems to replace (6.1.8) and (6.1.9) at certain places by the approximations ∂ ∼ ∂ = ∂X ∂x

and

∂ ∼ d . = ∂t dt

150

6 Linearization

Hutter and Pao, on the other hand, make full use of (6.1.8), but disregard (6.1.9) all together. This led to inconsistencies, which resulted in other ad hoc assumptions (as, for instance equation (5.7) in [91]). In the Eulerian formulation the linearization of jump conditions is equally tedious, because they are described at the deformed surface in the xstate, which ﬁrst must be traced back to the surface in the intermediate state. Corroboration for the fact that this procedure is rather cumbersome can be found e.g. in [91] or [249]. As we shall see in Sect. 5.2.4, in a Lagrangean description the linearization of jump conditions is straightforward. Several other authors have also published linearization procedures, which are all less general than that we shall present here, because these authors restrict themselves to (quasi-) magneto or electrostatic processes, to static intermediate states, or they simply delete jump conditions, etc. We conﬁne ourselves to mentioning Alblas [10], Pao and Yeh [171], Tiersten [233], Baumhauer and Tiersten [22] and Jordan and Eringen [104]. The advantage of the Lagrangean formulation is that the transformations (6.1.8) and (6.1.9) need not be applied, because in this formulation all operations are already referred to the reference or initial state. Similarly, the jump conditions hold on the undeformed surface. Once this is realized, the linearization procedure turns out to be straightforward. A second, but less direct advantage of the Lagrangean or material formulation is that it allows an immediate introduction of electromagnetic potentials for the perturbed electromagnetic ﬁelds, which is, although also possible, more elaborate in the Eulerian description. The reader may ﬁnd corroboration for this by noting (as we shall show lateron) that in the Lagrangean description the homogeneous Maxwell-equations (i.e. (2.2.15)) remain homogeneous in the perturbed state, whereas they become inhomogeneous in the Eulerian description (cf. [91], p. 81). This will then be our procedure. The nonlinear equations will be developed by consistently expanding all variables about the intermediate state and neglecting terms of order O(ε2 ). Two systems of equations emerge in doing so, one for the equations in the intermediate state and a second one for the perturbed quantities, in which the quantities of the intermediate state appear as coeﬃcients. From a practical point of view the construction of the solution of the perturbed equations is often more important than that of the intermediate state. Often the intermediate state serves only as a (static) biasing state deviations from which can be calculated by solving the perturbed equations. For buckling problems, for instance, the prebuckled state is taken as the intermediate state and its stability follows from the perturbed equations. Since, moreover, the perturbed equations are relatively insensitive to an exact determination of the intermediate ﬁelds, it would be advantageous to ﬁnd an approximation scheme by which the coeﬃcients could be determined to a suﬃcient degree of accuracy without making use of the exact intermediate equations. This is indeed often possible; we will demonstrate it

6.1 Statement of the Problem

151

below by introducing the so-called rigid-body state. The purpose of its use is to have a quick access to the perturbed equations without having to solve the equations in the intermediate state exactly. To explain it we consider the special situation where deformations, caused by the electromagnetic ﬁelds, are small and where changes in temperature are small as well. If at the same time coupling terms are small, which means that small deformations or small temperature changes result in small changes in the electromagnetic ﬁelds, then the intermediate state is close to an undeformed state. This imaginary state will be called rigid-body state. In this state the motion of the body, clearly, consists of a pure translation and rotation. The electromagnetic ﬁelds and the temperature distributions in this state are determinable from rigid-body electrodynamics, mechanics and thermodynamics. The ﬁelds in this state will be denoted by a superscript ( )0 , e.g. Bα0 , Eα0 , whereas subscripts ( )0 will indicate initial values prior to any motion. By mere deﬁnition we then have ρ0 = ρ0 ,

(6.1.10)

but generally neither Θ0 = Θ0 nor U 0 = 0. If U 0 = 0, then rigid-body motions are excluded. In most applications this will be the case. Since deformations are assumed to be small, we may deﬁne a measure E, 0 < E 1, by ¯α,β , (6.1.11) E := U with the aid of which we may conclude that ¯ − Θ0 | |Θ = O(E) , Θ0

¯ − B0 B = O(E) , || B 0

etc .

(6.1.12)

Returning to the perturbed equations, we notice that these equations are linear in the perturbed ﬁelds u, b, etc., with coeﬃcients, which depend on the values of the ﬁelds in the intermediate state. When the intermediate ﬁelds in these coeﬃcients are replaced by the rigid-body ﬁelds, errors of order O(E) are introduced. Because the perturbed equations are of order O(ε) themselves, the neglect of these terms ultimately means that the resulting equations are correct except for terms of order O(Eε). In short: The linearized perturbation equations, in which the coeﬃcients are referred to the rigid-body state, represent an approximation bound to an error not larger than O(Eε). It should further be emphasized that it is not justiﬁed, in general, to evaluate the intermediate ﬁelds themselves by applying the rigid-body approximations. This would result in errors of order O(E), which is larger than O(Eε). The approach just described to approximate the equations is more consistent than the usual small-strain approximations. In particular there are two immediate advantages: (i) the consistency of the linearization is a proven property and not an a priori assumption (the neglects can be made explicit) and,

152

6 Linearization

(ii) the stresses (and all other ﬁelds) in the intermediate state can still be calculated to within any desired degree of exactness. This latter point is of importance in particular for a consistent derivation of the equations governing magnetoelastic buckling of beams and plates (see [249], Ch. IX). As said above, we are still free to solve the equations in the intermediate state as accurately as we please. For most practical purposes, however, a small-strain approximation will suﬃce. We shall return to this point in due course with the developments in this chapter.

6.2 Linearization of the Lorentz Model 6.2.1 Motivation for this Choice – Governing Equations In this section we shall linearize the ﬁeld equations, constitutive relations and jump conditions of one particular interaction model, but before we present the details a justiﬁcation for our choice of the model is in order. We saw that all models are equivalent; as a consequence, only practical considerations and reasons of convenience can guide us to prefer one particular model over any other one. There are several reasons for the choice of model V. Firstly, the calculations in Chap. 5 have shown that the structure of many formulas is preserved when they are transformed from the spatial description into their material counterpart. Secondly, several thermodynamic relations, such as the relation between stress and free energy or the Gibbs relation, are much more consize for formulations with a symmetric than with an unsymmetric stress tensor. This would leave us with model II and model V, but model V is again computationally advantageous, because the Gauss–Faraday law and the Gauss law are homogeneous equations. This makes the introduction of electromagnetic potentials much easier than it would be otherwise. We shall deviate in one respect from the original Lorentz-model, however, in that we shall use QSα as energy-ﬂux instead of QL α . This essentially amounts to a diﬀerent choice of the entropy ﬂux (see Sects. 3.5 and 5.3).1 Moreover, we shall exclude external sources, so that ρ0 Fiext = ρ0 rext = 0 .

(6.2.1)

In the Lagrangean formulation and for the Lorentz model the governing equations read as follows

1

Compare the formulas (5.3.23) and (5.3.24).

6.2 Linearization of the Lorentz Model

153

Maxwell equations: (5.3.11) Bα,α = 0 , B˙ α + eαβγ Eγ,β = 0 , a = Q − Pα,α , Dα,α

(6.2.2)

a −D˙ αa + eαβγ Hγ,β = Jα + P˙ α + eαβγ ML γ,β ,

˙ + Jα,α = 0 , Q where

−1 Dαa = ε0 |J|Cαβ Eβ ,

1 1 a Cαβ Bβ − ε0 eµβγ Cαβ Fjγ x˙ j Eµ . Hα = |J| µ0

Balance of momentum: ((5.6.9)1 , with Fiext = 0) −1 (Q − Pβ,β )Eα + eαβγ (Jβ + P˙ β )Bγ ¨i − Tiα,α = ρ0 Fie = Fαi ρ0 x L . +(ML − M )B β α,β β,α

(6.2.3)

(6.2.4)

Balance of energy: ((5.6.9)3 , with Qα = QSα and rext = 0) ρ0 Θη˙ = Jα Eα − Qα,α .

(6.2.5)

In the above, all variables are referred to the time-independent reference state characterized by the coordinates Xα . Superimposed dots thus represent time derivatives at ﬁxed particles. Constitutive Relations: In accordance with (5.3.35) we choose as independent variables Cαβ , Eα , Bα , Θ, Θ,α and Q . Following (5.3.36) and (5.3.37), we then have 1 ˇ αβ , Eα , Bα , Θ) , ψˇ = U − Θη + Eα Pα = ψ(C ρ0 and η=− ML α

∂ ψˇ , ∂Θ

∂ ψˇ = −ρ0 , ∂Bα

∂ ψˇ , ∂Eα ∂ ψˇ = 2ρ0 . ∂Cαβ

(6.2.6)

Pα = −ρ P Tαβ

(6.2.7)

154

6 Linearization

Moreover, for an electrical conductor the constitutive relations for the electric current and the energy ﬂux are of the form (see (5.5.12) with (5.5.14)3 ) Jα = σαβ Eβ + ββα

Θ,β , Θ

Qα = −καβ Θ,β + βαβ Eβ ,

(6.2.8)

where, in general, the coeﬃcient matrices are still functions of Cαβ , Eα , Bα , Θ, Θ,α and Q. However, in accordance with the linearization as described in the derivation of (3.7.34) we shall assume these coeﬃcients to be independent of Eα and Θ,α . Moreover, we shall exclude an explicit occurrence of Q. Equations (6.2.8) automatically guarantee that the current Jα and heat ﬂux Qα vanish in thermostatic equilibrium; in short, if Eα = 0 and Θ,α = 0, then and Qα|E = 0 . Jα|E = 0 Further conditions of thermostatic equilibrium are that σ(αβ) and −κ(αβ) must be positive-semi deﬁnite matrices and, if the Onsager relations are adopted, that the skew symmetric parts of σαβ and καβ must vanish. Jump conditions: ((5.3.15), (2.4.12), (5.3.25)) Bα Nα = 0 , eαβγ Eβ Nγ + Bα WN = 0 , a Dα + Pα Nα = 0 , a eαβγ Hβa − ML β Nγ − (Dα + Pα )WN = 0 , Jα Nα − QWN = 0 , ρ0 WN = 0 , ρ0 x˙ i WN + Tiα + TM iα Nα = 0 , 1 ( 2 ρ0 x˙ i x˙ i + ρ0 U − Ω)WN + (Tiα + TM ˙ i − Qα − eαβγ Eβ (Haγ − ML γ ) Nα = 0 . iα )x

(6.2.9)

In these relations, Nα is the unit normal vector on the singular surface, WN M is the speed of propagation, and Tiα and Ω are given in (5.3.25); for ease of reference they will be repeated here: −1 M Tiα = Fβi Dαa Eβ + Bα Hβa − 12 δαβ (Dγa Eγ + Bγ Hγa ) , (6.2.10) Ω = − 12 (Dαa Eα + Bα Hαa ) . In this chapter we shall from now on restrict ourselves to singular surfaces of second order (see e.g. [68], Sect. 2.8). On such a surface, the deformation gradients and the velocity will be continuous, as is the density ρ0 , i.e. Fiα = x˙ i = ρ0 = 0 .

6.2 Linearization of the Lorentz Model

155

In view of (6.2.9)6 continuity of ρ0 also implies that the speed of propagation WN is continuous, WN = 0 . For the case that the singular surface is a real propagating surface, i.e. WN = 0, the associated waves are called acceleration waves. Singular surfaces of second order include as a special case material surfaces for which WN = 0, and also embrace the boundary of a solid body in a vacuum, if, as is usually the case, the vacuum is considered as a medium with zero density, which admits continuity of the velocities at the boundary. Surfaces on which tangential velocities may jump are, however, excluded. Under the above restrictions the last two jump conditions can be simpliﬁed. In view of the continuity of ρ0 , x˙ i and WN , (6.2.9)7 becomes M Tiα Nα = − Tiα Nα . (6.2.11) This equation can be written in the form L −1 Tiα Nα = Fγi Eγ Pβ Nβ + Bβ ( ML β N γ − Mγ N β ) ( + eβγδ Bβ Pδ WN ,

(6.2.12)

in which the symbol · stands for the arithmetic mean of a quantity over the singular surface, i.e. − (6.2.13) Eα = 12 (E+ α + Eα ) . To prove (6.2.12), notice that in view of relations (6.2.3) and of the continuity conditions of second-order singular surfaces, Hαa Bα = Bα Hαa . (6.2.14) Dαa Eα = Eα Dαa , Moreover, the jump of the ﬁrst term in the outer brackets of (6.2.10)1 may be written as a Dα Eβ Nα = (Dαa Eβ + Dαa Eβ )Nα = Dαa Nα Eβ − Eβ Pα Nα , in which use has been made of (6.2.9)3 . With (6.2.14)1 and (6.2.9)4 it is then straightforward to show that in the non-relativistic approximation a Dα Eβ Nα − 12 Dγa Eγ δαβ Nα = −Eβ Pα Nα + eαβγ Dαa Bγ WN = −Eβ Pα Nα . In an analogous way the remaining terms on the right-hand side of (6.2.10), (6.2.11) can be handled, so that, at last, (6.2.12) emerges. On the other hand, the ﬁrst and fourth term in (6.2.9)8 vanish for acceleration waves; furthermore, with the use of (6.2.9)2,4 it can be shown that

156

6 Linearization

eαβγ Eβ (Hγa − ML γ ) Nα = a Bα Hαa − ML WN . + E D + P α α α α

(6.2.15)

Finally, from (6.2.10)2 and with the application of relations (6.2.14) we ﬁnd Ω = −Eα Dαa − Hαa Bα . (6.2.16) With these results the jump condition for the energy ﬂux assumes the simple and elegant form Qα Nα = ρ0 U + ML − E B Pα WN . (6.2.17) α α α It should be noted here that the right-hand side vanishes when the surface is material. Before we close, consider a surface separating a body from the vacuum. According to our interpretation of the vacuum as a medium with vanishingly small mass density, and because we are looking at surfaces of second order, the deformation tensor is continuous across such a surface so that on the vacuum side (+) one has −1 + + −1 − + ) Eβ = (ε0 JCαβ ) Eβ . Dαa+ = (ε0 JCαβ + Similar statements also hold for Hαa+ . Note also that E+ α = Ei δiα ; one rather has − + + ˙− E+ α = Fiα (Ei + eijk x j Bk )

and

−1 − + B+ α = (JFαi ) Bi .

These facts should be borne in mind, for otherwise incorrect results emerge. This completes the collection of the basic governing equations. Their linearized versions will be derived below. 6.2.2 Decomposition of the Balance Laws Having presented the governing equations we now proceed with the decomposition of the balance laws. As was said several times before already, this decomposition is particularly easy in the material description. To corroborate this statement, let us consider the ﬁrst of the Maxwell equations (6.2.2). Introducing (6.1.3) as decomposition for the magnetic induction Bα , we may write equation (6.2.2)1 as ¯ α,α + bα,α = 0 . ¯ α + bα ),α = B (B

(6.2.18)

In view of the basic separation assumption, according to which the governing equations must be fulﬁlled for the intermediate state itself, we thus have

6.2 Linearization of the Lorentz Model

157

¯ α,α = 0 , B

(6.2.19)

bα,α = 0 .

(6.2.20)

and, consequently, ¯ α and bα are regarded as functions of Xα and t. In these equations, both B Proceeding with all Maxwell equations as was explained above with the Gauss law, we can easily show that the equations, valid in the intermediate state, are given by ¯ α,α = 0 , B

¯˙ α + eαβγ E ¯γ,β = 0 , B

¯a =Q ¯ −P ¯ α,α , D α,α

¯˙ a + eαβγ H ¯˙ α + eαβγ M ¯a =¯ ¯L , −D Jα + P α γ,β γ,β

(6.2.21)

¯˙ + ¯ Q Jα,α = 0 . These equations are obtained from the original Maxwell equations (6.2.2) by merely replacing in the latter all quantities by those carrying an overhead bar. On the other hand, the perturbed Maxwell equations are2 bα,α = 0 ,

b˙ α + eαβγ eγ,β = 0 ,

daα,α = q − pα,α ,

−d˙ aα + eαβγ haγ,β = jα + p˙ α + eαβγ mγ,β ,

(6.2.22)

q˙ + jα,α = 0 , in which all lower-case letters denote perturbed Lagrangean electromagnetic ﬁeld quantities, the deﬁnitions being analogous to that for bα . For instance, ¯ α , etc. eα = Eα − E For convenience and since no confusion is possible here, we have used mα L ¯L for (ML α − Mα ) instead of mα . Whenever, in the sequel confusion with the Chu-magnetization becomes possible we shall use mL α , however. Equations (6.2.22) are formally the same as the original Maxwell equations. This is no surprise, because equations (6.2.2)1−5 are written such that they appear in a linear form. However, they are nevertheless nonlinear; the nonlinearity is only covered by the use of the auxiliary ﬁelds Dαa and Hαa . In the decomposition process of these quantities use must be made of the following relations, of which the proof is straightforward: 2

In this chapter the perturbed ﬁelds from the intermediate state will be denoted by Roman letters and not by italics. Alternatively, ﬁelds in the intermediate state are identiﬁed by an overhead bar. This notation has already been used in (6.2.18)–(6.2.20).

158

6 Linearization

xi = δiα (ξα + uα ) , ∂ξβ ∂xi ∂uβ Fiα = = δiβ + = δiβ (F¯βα + uβ,α ) , ∂Xα ∂Xα ∂Xα ∂Xα ∂Xα ∂ξβ ∼ ¯ −1 −1 −1 = = uβ,γ ) , Fαi = Fαβ (δiβ − δiδ F¯γδ ∂xi ∂ξβ ∂xi Cαβ = Fiα Fiβ ∼ = C¯αβ + F¯γβ uγ,α + F¯γα uγ,β ,

(6.2.23)

−1 −1 −1 ∼ ¯ −1 −1 ¯ −1 −1 ¯ −1 Fαδ + C¯αγ Fβδ )uδ,γ , Cαβ = Fαi Fβi = Cαβ − (C¯βγ −1 ¯ + F¯ −1 uδ,γ ) . J = det(Fiα ) = det[F¯βα (δiβ + δiδ F¯βγ uδ,γ )] ∼ = J(1 γδ

Here, ∂ξα ∂Xα −1 F¯αβ = , F¯αβ = , C¯αβ = F¯γα F¯γβ , ∂Xβ ∂ξβ −1 −1 ¯ −1 Fβγ , C¯αβ = F¯αγ J¯ = det(F¯αβ ) .

(6.2.24)

With these preliminary calculations the decomposition of the auxiliary ﬁelds Dαa and Hαa can now be performed. The results are ¯ a = ε0 J¯C¯ −1 E ¯ D α αβ β , ¯a= H α and

1 ¯ ¯ ε0 ¯δ , Cαβ Bβ − ¯ eβγδ C¯αβ F¯εγ ξ˙ε E ¯ J µ0 J

(6.2.25)

−1 −1 ¯ −1 −1 ¯ −1 −1 ¯ −1 ¯ daα = ε0 J¯ C¯αβ Fγδ − C¯βγ Fαδ − C¯αγ Fβδ )Eβ uδ,γ , eβ + (C¯αβ " 1 1 ¯ a Cαβ bβ − ε0 eβγδ F¯εγ ξ˙ε C¯αβ eδ hα = ¯ J µ0 1 ¯ ¯µ F¯δα δβγ + F¯δβ δαγ − C¯αβ F¯γδ + Bβ − eβνµ F¯εν ξ˙ε E µ0 ! # ˙ ¯ ¯ ¯ ¯ ¯ −ε0 eβγµ Eµ Cαβ ξδ uδ,γ − ε0 eβγδ Eδ Cαβ Fεγ u˙ ε .

(6.2.26) Here and henceforth we restrict ourselves to positive values of the Jacobian J and could therefore replace |J| by J whenever it occurred. We further would like to emphasize that the perturbed ﬁelds daα and haα are expressed here in terms of the deformation and the electromagnetic ﬁelds in the intermediate state, all of which are functions of Xα and t, in general. The deformation in the intermediate state may, in our approximation, be neglected and the intermediate ﬁelds be replaced by the ﬁelds in the rigidbody state. If this is the case we may choose ξα0 (X, t) = Ξα (t) + Rαβ (t)Xβ ,

(6.2.27)

6.2 Linearization of the Lorentz Model

159

where Ξα (t) can be identiﬁed with the coordinates of the center of mass of the body in its rigid-body state (the position of the center of mass in the reference state, t = 0 is here taken as the origin of our coordinate system), and where Rαβ (t) is a time-dependent proper orthogonal matrix, i.e. R−1 = RT

and

det R = +1 .

Diﬀerentiating both sides of (6.2.27) with respect to time, we obtain ξ˙α0 = Ξ˙ α + R˙ αβ Xβ = Ξ˙ α + eαβγ Ωβ Rγδ Xδ ,

(6.2.28)

where Ω(t) is the angular velocity of the body, which is given by Ωα = − 12 eαβγ R˙ βδ Rγδ .

(6.2.29)

The approximate versions of equations (6.2.26) can now be obtained by replacing in these equations all variables carrying an overhead bar, (¯·), by the variables in the rigid-body state, (·)0 . In other words, the following replacements must be made ¯α → E 0 , E α J¯ → 1 ,

¯α → B 0 , B α C¯αβ → δαβ ,

F¯αβ → Rαβ ,

−1 F¯αβ → Rβα ,

−1 C¯αβ → δαβ .

(6.2.30)

With these we obtain for (6.2.26) daα = ε0 eα + (δαβ Rδγ − δβγ Rδα − δαγ Rδβ )Eβ0 uδ,γ , haα =

1 bα − ε0 eαβγ Rδβ ξ˙δ0 eγ µ0 1 0 + Bβ − eβνµ Rεν ξ˙ε0 Eµ0 )(Rδα δβγ + Rδβ δαγ − Rγδ δαβ µ0 ! −ε0 eαγµ E 0 ξ˙0 uδ,γ − ε0 eαβγ E 0 Rδβ u˙ δ . µ δ

(6.2.31)

γ

Still further simpliﬁcations can be made, if the intermediate state is not only close to a rigid-body state, but deviates from the initial state by a small amount only (i.e. for a motionless rigid-body state). In that case equations (6.2.31) are further simpliﬁed by setting Rαβ = δαβ

and

ξ˙α0 = 0 .

The next step in the simpliﬁcation of the balance laws is the decomposition of the momentum and energy equations. To this end, (6.2.4) and (6.2.5) will be written as ¨α )δiα = T¯iα,α + δiα tαβ,β + ρ0 F¯ie + ρ0 fαe δiα , (ρ0 ξ¨α + ρ0 u ¯ + θ)(η¯˙ + s˙ ) = (Jα + jα )(Eα + eα ) − (Q ¯ α − qα ),α . ρ0 (Θ

(6.2.32)

160

6 Linearization

Apart from the perturbation variables already introduced before we have deﬁned here fαe = δiα (Fie − F¯ie ) , s = η − η¯ ,

tαβ = δiα (Tiβ − T¯iβ ) ,

¯, θ =Θ−Θ

¯α . qα = Qα − Q

(6.2.33)

Because we assume that equations (6.2.32) must hold for the intermediate state, we have ρ0 ξ¨α = δiα (T¯iβ,β + ρ0 F¯ie ) , (6.2.34) ¯α − Q ¯ η¯˙ = ¯ ¯ α,α , Jα E ρ0 Θ as momentum and energy equations in this state and ρ0 u ¨α = tαβ,β + ρ0 fαe , ¯α + ¯ ¯ s ) = jα E Jα eα − qα,α , ρ0 (θη¯˙ + Θ˙

(6.2.35)

as the corresponding equations in the perturbed state. An explicit representation of the body force expression is obtained from (6.2.4) and (6.2.23)3 , namely −1 ¯ −P ¯ β,β )E ¯ α + eαβγ (¯ ¯˙ β )B ¯ γ + (M ¯L −M ¯ L )B ¯β} , ρ0 F¯ie = δiδ F¯αδ {(Q Jβ + P α,β β,α (6.2.36) as the expression for the body force in the intermediate state, and ¯ −P ¯ β,β )eδ + (q − pβ,β )E ¯ε ρ0 f e = F¯ −1 (Q α

δα

¯˙ β )bγ + (jβ + p˙ β )B ¯γ +eβγδ (Jβ + P

¯ β (mδ,β − mβ,δ ) + bβ (ML − ML ) +B δ,β β,δ ¯ ¯ ¯˙ ¯ ¯ −1 ¯ ¯ ¯ −1 F −F βγ δα (Q − Pε,ε )Eβ + eβµν (Jµ + Pµ )Bν ¯ ε (M ¯L −M ¯ L ) uγ,δ , +B β,ε ε,β

(6.2.37)

as the corresponding expression in the perturbed state. When deformations in ¯ αβ the intermediate state are neglected, but rigid motions are allowed, then F in (6.2.37) must simply be replaced by Rαβ and, furthermore, all variables carrying an overhead bar must be replaced by the variables in the rigid-body state. Moreover, for a motionless rigid-body state Rαβ = δαβ and ξ˙α0 = 0. The balance laws of mechanics and electrodynamics are thus decomposed into equations valid in the intermediate state and those in the perturbed state. When in the latter the rigid-body approximation is applied, the corresponding equations in this state are also needed. The governing equations consist of the Maxwell equations

6.2 Linearization of the Lorentz Model

161

0 Bα,α =0,

0 B˙ α0 + eαβγ Eγ,β =0,

a0 0 = Q 0 − Pα,α , Dα,α

a0 −D˙ αa0 + eαβγ Hγ,β = J0α + P˙ α0 + eαβγ ML0 γ,β , (6.2.38)

˙ 0 + J0 = 0 , Q α,α where Dαa0 = ε0 Eα0 ,

Hαa0 =

1 0 B − ε0 eαβγ Rδβ ξ˙δ0 Eγ0 , µ0 α

(6.2.39)

and of the energy equation ρ0 Θ0 η˙ 0 = J0α Eα0 − Q0α,α ,

(6.2.40)

supplemented by the equations for the rigid-body motion (which will be stated below), by the pertinent constitutive relations and by the jump conditions. Equations (6.2.38)–(6.2.40), together with constitutive relations and jump conditions, suﬃce to determine the rigid-body ﬁelds, provided that the rigidbody state is motionless (ξ˙α0 = 0) or has a prescribed motion. If this is not the case, the motion (i.e. Ξα (t) and Rαβ (t) or Ωα (t)) must be determined from momentum equations, which can best be handled, if the balance laws of linear and angular momentum are recast into global form. We restrict ourselves to a rigid body placed in a vacuum, in which case these relations are d M 0 ρ0 ξ˙α0 dV = δiα (Tiβ + Tiβ ) Nβ dA + ρ0 Fαext dV , dt Ω Ω ∂Ω − d 0 0 0 M 0 ˙ ρ0 (ξ[α − Ξ[α )ξβ] dV = (ξ[α − Ξ[α )(Tiγ + Tiγ ) δiβ] Nγ dA (6.2.41) dt − Ω ∂Ω 0 ext + ρ0 {Lext αβ + (ξ[α − Ξ[α )Fβ] }dV . Ω M is the Maxwell stress tensor as deﬁned in (6.2.10)1 and Fαext Here, Tiα ext and Lαβ are externally applied body forces and body couples with respect to the center of mass, respectively. Ω is the body manifold and ∂Ω − the surface of Ω just inside the boundary of the body. According to the jump condition (6.2.11), which is applied for the boundary of the body separating the vacuum, it is possible to simplify the surface terms in (6.2.41), for (N )

M 0 ) Nβ )δΩ − = (ti ((Tiβ + Tiβ (N )

M 0 + (Tiβ ) Nβ )∂Ω + ,

are the surface tractions of other than electromagnetic origin, where ti M )∂Ω + is the Maxwell stress tensor evaluated just outside the boundary (Tiβ of the body and where ∂Ω + represents a surface just outside the boundary of the body.

162

6 Linearization

With the aid of (6.2.28) we may then transform (6.2.41) into MΞ¨α = Fαe + Fαext , d (Iαβ Ωβ ) = Leα + Lext α , dt where

ρ0 dV ,

M=

Iαβ =

Ω

ρ0 δαβ Xγ Xγ − Rαγ Rβδ Xγ Xδ dV ,

(6.2.42)

(6.2.43)

Ω

are the total mass and the instantaneous central moments of inertia of the body, M 0 e δiα (Tiβ ) Nβ dA , Fα =

∂Ω +

Leα

= eαβγ δiγ

M 0 (ξβ0 − Ξβ )(Tiδ ) Nδ dA ,

(6.2.44)

∂Ω +

are the resultant force and moment of electromagnetic origin relative to the center of mass, and where (N ) Fαext = δiα ti dA + ρ0 Fαext dV ,

∂Ω +

Leα

Ω

(ξβ0

= eαβγ

(N )

− Ξβ )ti

δiγ dA

(6.2.45)

∂Ω + 0 ext ρ0 {Lext βγ + (ξβ − Ξβ )Fγ }dV ,

+eαβγ Ω

are the resultant force and moment of the externally applied forces relative to the center of mass. To summarize, the velocity of the center of mass Ξ˙ α and the angular velocity Ωα of a rigid body moving in an electromagnetic ﬁeld are governed by equations (6.2.42), in which the electromagnetic surface forces and surface couples acting on the body are determined from the Maxwell stress tensor in the surrounding vacuum. Clearly, equations (6.2.42) must be solved along with the Maxwell equations and the energy equation in the rigid-body state, (6.2.38)–(6.2.40). In these equations the angular velocity Ωα and the matrix Rαβ arise; these themselves are connected by equation (6.2.29). This completes the decomposition of the electromagnetic and mechanical balance laws. In the following sections the decomposed versions of the constitutive relations and jump conditions will be given. With these the zerothorder solution can be solved, and once this is done the ﬁrst-order perturbed problem can be attacked.

6.2 Linearization of the Lorentz Model

163

6.2.3 Decomposition of the Constitutive Equations The constitutive equations for entropy, polarization, magnetization and stress are known, once the functional ψˇ as deﬁned in (6.2.6) is speciﬁed. In order to make the theory complete, constitutive equations for Qα and Jα must also be established. Assuming ψˇ to be diﬀerentiable, we may expand it in terms of Taylor series about the intermediate state (eventually approximated by the rigid-body state). The series is truncated and substituted into (6.2.7) and what emerges are expressions for entropy, polarization, magnetization and the Piola–Kirchhoff stress tensor, which all are linear in the perturbed quantities eα , bα , uα , etc. The expansion of the constitutive equations into Taylor series is straightforward and could be performed formally without ˇ Such expressions are of little use, howspecifying the energy functional ψ. ever, because ultimately one must specify the free energy anyhow. A common procedure, applicable for many practical purposes, is to assume ψˇ to be a polynomial in its independent variables, which can be truncated at a certain order. ˇ which contains all interaction eﬀects at A reasonable expression for ψ, least to within ﬁrst-order terms is 1 1 1 (m) (m) (e) Eα Eβ ψˇ = 2 χαβ Bα Bβ + 4 χαβγδ Bα Bβ Bγ Bδ + 2χ 2ρ0 4ρ0 2ρ0 αβ 1 1 (em) (e) + Eα Eβ Eγ Eδ + χαβ Bα Eβ − 12 c(Θ − Θ0 )2 4χ 4ρ0 αβγδ ρ0 1 1 (m) + λ(m) Bα (Θ − Θ0 ) + L Bα Bβ (Θ − Θ0 ) ρ0 α 2ρ0 αβ 1 (e) 1 L Eα Eβ (Θ − Θ0 ) + λ(e) α Eα (Θ − Θ0 ) + ρ0 2ρ0 αβ 1 (m) (m) (e) (e) ε Bβ + 12 bαβγδ Bα Bβ + εβγδ Eβ + 12 bαβγδ Eα Eβ + ρ0 βγδ 1 −ρ0 νγδ (Θ − Θ0 ) Eγδ + cαβγδ Eαβ Eγδ . 2ρ0

(6.2.46)

Here, instead of Cαβ we have used Eαβ := 12 (Cαβ − δαβ ) ,

(6.2.47)

as deformation measure. According to (6.2.23)4 this can be developed as ¯αβ + eαβ , Eαβ = E

(6.2.48)

where ¯αβ = 1 (C¯αβ − δαβ ) , E 2

eαβ = 12 (F¯γβ uγ,α + F¯γα uγ,β ) .

(6.2.49)

Neglecting the deformations in the ξ-state, we may approximate the latter by

164

6 Linearization

eαβ = 12 (Rγβ uγ,α + Rγα uγ,β ) .

(6.2.50)

The polynomial representation (6.2.46) for the free energy is such that elastic and thermal eﬀects are essentially linear and that interactions with three ﬁelds are excluded. Yet, some magnetic and electric nonlinearities (fourth order terms) are nevertheless present. The reasons for this are twofold: (i) For ferromagnetic materials and for cubic crystals fourth-order anisotropy (m) eﬀects (represented by 4 χαβγδ ) are often rather important and may, frequently, even be more important than the second-order terms; for cubic crystals fourth-order terms are the only nontrivial magnetic anisotropy eﬀects, in general. (ii) For materials with central symmetry (e.g. cubic crystals or isotropic bodies) ﬁrst- and third-order coeﬃcients in (6.2.46) vanish. In this case second- and fourth-order terms such as Lαβ and bαβγδ must be dominant. We further note that it is possible to extend the above functional form of the free energy to include still higher-order eﬀects without the result that this would fundamentally inﬂuence the subsequent analysis. This will not be done here, and for the time being we also refrain from assigning names to the various coeﬃcients occurring in (6.2.46), simply because the results of Sect. 5.7 have shown that unique interpretations are not possible. The coeﬃcients occurring in (6.2.46) also satisfy certain symmetry requirements, which are readily obtained from (6.2.46). Because they are so obvious, we shall not explicitly state them here. It is now straightforward to derive explicit expressions for entropy, polarization, magnetization and stress by substituting (6.2.46) into (6.2.7) and performing the respective diﬀerentiations. For the entropy this leads to 1 (m) 1 (m) λ Bα − L Bα Bβ ρ0 α 2ρ0 αβ 1 1 (e) − λ(e) Eα − L Eα Eβ + ναβ Eαβ ρ0 α 2ρ0 αβ 1 (m) ¯ ¯ 1 ¯ ¯α ¯ − Θ0 ) − 1 λ(m) = c(Θ Lαβ Bα Bβ − λ(e) E α Bα − ρ0 2ρ0 ρ0 α 1 (e) ¯ ¯ ¯ ¯αβ + cΘ − 1 (λ(m) + L(m) B − L Eα Eβ + ναβ E αβ α )bα 2ρ0 αβ ρ0 α 1 (e) ¯ − (λ(e) + Lαβ E ¯+s . β )eβ + ναβ eαβ =: η ρ0 α

η = +c(Θ − Θ0 ) −

(6.2.51)

Here, O(ε2 )-terms are neglected and the entropy of the intermediate state is given by 1 (e) ¯ ¯ α − 1 L(m) B ¯ ¯ ¯ − Θ0 ) − 1 λ(m) B η¯ = c(Θ λ Eα α αβ α Bβ − ρ0 2ρ0 ρ0 α 1 (e) ¯ ¯ ¯αβ , − L Eα Eβ + ναβ E 2ρ0 αβ

(6.2.52)

6.2 Linearization of the Lorentz Model

165

whilst the perturbed entropy s may be written as s = s1αβ uα,β + s2α bα + s3α eα + s4 θ ,

(6.2.53)

with coeﬃcients, which can easily be derived from (6.2.51) and (6.2.49)2 as s1αβ = F¯αγ νβγ , s3α = −

s2α = −

1 (m) (m) ¯ (λ + Lαβ B β) ρ0 α

1 (e) (e) ¯ (λ + Lαβ E β) , ρ0 α

(6.2.54)

s4 = c .

When deformations of the intermediate state are neglected we simply set F¯αβ = Rαβ

(¯·) = (·)0 .

and

In a completely analogous way also the constitutive equations for polarization, magnetization, and Piola–Kirchhoff stress can be deduced. One obtains for the • polarization ¯ α + pα = P ¯ α + p1 uβ,γ + p2 bβ + p3 eβ + p4 θ , Pα = P αβγ αβ αβ α

(6.2.55)

where (e) (e) ¯ ¯ ¯ ¯ ¯ ¯ α = −(ε(e) + b(e) E P αγδ αβγδ β )Eγδ − ( 2 χαβ + 4 χαβγδ Eγ Eδ )Eβ , (em) ¯ (e) (e) ¯ ¯ −χβα B β − (λα + Lαβ Eβ )(Θ − Θ0 ) ,

(6.2.56)

and (e)

(e)

¯ ε) , p1αβγ = −F¯βδ (εαγδ + bαεγδ E (em)

p2αβ = −χβα , (e)

(e)

(e)

(e)

(6.2.57)

¯γE ¯δ − b ¯ ¯ p3αβ = − 2 χαβ − 3 4 χαβγδ E αβγδ Eγδ − Lαβ (Θ − Θ0 ) , (e)

(e)

¯β; p4αβ = −λα − Lαβ E • magnetization ¯ α + mα = M ¯ α + m1 uβ,γ + m2 bβ + m3 eβ + m4 θ , (6.2.58) Mα = M αβγ αβ αβ α where (m) (m) ¯ ¯ ¯ ¯ ¯ ¯ α = −(ε(m) + b(m) B M αγδ αβγδ β )Eγδ − ( 2 χαβ + 4 χαβγδ Bγ Bδ )Bβ , (em) ¯ (m) (m) ¯ ¯ −χαβ E β − (λα + Lαβ Bβ )(Θ − Θ0 ) ,

and

(6.2.59)

166

6 Linearization (m) (m) ¯ m1αβγ = −F¯βδ (εαγδ + bαεγδ B ε) , (m)

(m)

(m)

(m)

¯γB ¯δ − b ¯ ¯ m2αβ = − 2 χαβ − 3 4 χαβγδ B αβγδ Eγδ − Lαβ (Θ − Θ0 ) , (em)

m3αβ = −χαβ , m4α

(m)

= −λα

(m) ¯ − Lαβ B β .

(6.2.60) • Piola–Kirchhoff stress tensor, which according to (6.2.7)4 and (5.2.24) is given by ∂ ψˇ Fiγ , (6.2.61) Tiβ = ρ0 ∂Eβγ so that Tiβ = T¯iβ + δiα tαβ = T¯iβ + δiα (t1αβγδ uγ,δ + t2αβγ bγ + t3αβγ eγ + t4αβ θ) , (6.2.62) where ¯ ¯ ¯δε + (ε(m) + 1 b(m) B T¯iβ = δiα F¯αγ [cβγδε E δβγ 2 δεβγ ε )Bδ (e) (e) ¯ ¯ ¯ +(εδβγ + 12 bδεβγ E ε )Eδ − ρ0 νβγ (Θ − Θ0 )] ,

(6.2.63)

and ¯ ¯ ¯µν + (ε(m) + 1 b(m) B t1αβγδ = F¯αµ F¯γε cβµεδ + δαγ [cβδµν E εβδ 2 εµβδ µ )Bε (e) (e) ¯ ¯ ¯ +(εεβδ + 12 bεµβδ E µ )Eε − ρ0 νβδ (Θ − Θ0 )] , (m) (m) ¯ t2αβγ = F¯αδ (εγβδ + bγεβδ B ε) ,

(6.2.64)

(e) (e) ¯ t3αβγ = F¯αδ (εγβδ + bγεβδ E ε) ,

t4αβ

= −ρ0 F¯αγ νβγ .

The above derivation is perfectly general and applies whether deformations in the intermediate state are small or large. If they are small, all coeﬃcients characterized by a lower case letter (say p1αβγ , t4αβ etc.) may further be simpliﬁed by replacing all quantities with an overhead bar by the corresponding rigid-body quantities. The error introduced into the balance laws of the perturbed quantities is then of order O(Eε). Note that the term containing cβµεδ in the expression for t1αβγδ is, in practice, always much larger than the remaining terms. This property is the justiﬁcation for the assumption that the deformations due to the electromagnetic and due to the thermal ﬁelds are small. It seems to be reasonable, therefore, to approximate t1αβγδ by t1αβγδ ∼ = Rαµ Rγε cβµεδ .

(6.2.65)

6.2 Linearization of the Lorentz Model

167

Here, we have written Rαµ for F¯αµ because the error introduced by dropping the remaining terms in (6.2.64)1 is of the same order as that obtained by the replacement F¯αβ → Rαβ . We conclude this section with the presentation of the decomposition of the generalized versions of Ohm’s and Fourier’s law, as given in (6.2.8). The coeﬃcients in these equations are functions of the variables Eαβ , Eα , Bα , Θ and Θ,α . We shall exclude a possible dependence of these constitutive relations on the free charge Q, and shall, furthermore, restrict ourselves to a linear dependence of current and heat ﬂux on electric ﬁeld and temperature gradient. We then have (6.2.66) Λαβ = Λαβ (Eγδ , Bγ , Θ) , where Λαβ stands for σαβ , καβ and βαβ , respectively. To be more speciﬁc we shall choose the following polynomial expansions (r) (d) (m) (m) (t) ¯ − Θ0 ) . (6.2.67) Λαβ = Λαβ + Λαβγδ Eγδ + 3 Λαβγ Bγ + 4 Λαβγδ Bγ Bδ + Λαβ (Θ

Here we have included second-order terms in Bα , but have deleted thirdorder terms, because they vanish in a material with point symmetry anyhow. Decomposing the representations (6.2.8) for electric current and energy ﬂux, we ﬁnd ¯ ¯ β + β¯βα Θ,β , ¯ ¯αβ E Jα = σ ¯ (6.2.68) Θ ¯ ¯β ¯ ¯ Qα = −¯ καβ Θ,β + βαβ E as governing equations in the intermediate state and jα = j1αβγ uβ,γ + j2αβ bβ + j3αβ eβ + j4α θ + j5αβ θ,β , qα = q1αβγ uβ,γ + q2αβ bβ + q3αβ eβ + q4α θ + q5αβ θ,β

(6.2.69)

as the approximate equations in the perturbed state, in which ¯ (d) ¯ (d) Θ,ε j1αβγ = F¯βδ (σαεγδ E ε + βεαγδ ¯ ) , Θ ¯ (m) (m) ¯δ E ¯ Θ,γ , ¯ γ + 3 β (m) + 2 4 β (m) B j2αβ = 3 σαγβ + 2 4 σαγβδ B γαβ γαβδ δ ¯ Θ (r) (d) ¯ (m) ¯ (m) ¯ ¯ (t) ¯ 3 j Eγδ + 3 σ =σ +σ Bγ + 4 σ Bγ Bδ + σ (Θ − Θ0 ) , αβ

j4α

j5αβ

αβ

αβγδ

αβγ

αβγδ

(r) (d) ¯ (m) ¯ (m) ¯ ¯ = − ββα + ββαγδ E γδ + 3 ββαγ Bγ + 4 ββαγδ Bγ Bδ

αβ

Θ ¯ ,β ¯ (t) ¯ (t) ¯ (t) Θ,β +ββα (Θ − Θ0 ) ¯ 2 + σαβ E β + ββα ¯ , Θ Θ 1 (r) (d) ¯ (m) ¯ (m) ¯ ¯ (t) ¯ = ¯ ββα + ββαγδ Eγδ + 3 ββαγ Bγ + 4 ββαγδ B + β ( Θ − Θ ) , B γ δ 0 βα Θ (6.2.70)

168

6 Linearization

and (d) ¯ (d) ¯ q1αβγ = F¯βδ (−καεγδ Θ ,ε + βαεγδ Eε ) , (m) (m) ¯ (m) (m) ¯ ¯ ¯ q2αβ = − 3 καγβ + 2 4 καγβδ B δ Θ,γ + 3 βαγβ + 2 4 βαγβδ Bδ Eγ , (r) (d) ¯ (m) ¯ (m) ¯ ¯ (t) ¯ q3αβ = βαβ + βαβγδ E γδ + 3 βαβγ Bγ + 4 βαβγδ Bγ Bδ + βαβ (Θ − Θ0 ) , (t) ¯ (t) ¯ = −καβ Θ ,β + βαβ Eβ , (r) (d) ¯ γ + 4 κ(m) B ¯γB ¯ δ + κ(t) (Θ ¯γδ + 3 κ(m) B ¯ − Θ0 ) . q5αβ = − καβ + καβγδ E αβγ αβγδ αβ (6.2.71) Speciﬁc forms of the coeﬃcients, however only for isotropic materials, have been derived by Pipkin and Rivlin, [180, 181], and by Borghesani and Morro, [27, 28]. These authors also discuss the physical signiﬁcance of the various terms in (6.2.70) and (6.2.71). At this point we have thus completed the presentation of the linearized ﬁeld equations. If (6.2.53), (6.2.55), (6.2.58), (6.2.62) and (6.2.69) are substituted into the balance laws (6.2.22) and (6.2.35), what we obtain is a system of 11 independent ﬁeld equations for the 11 unknowns bα , eα , q, uα and θ. This system could still further by simpliﬁed by introducing electromagnetic potentials ϕ and aα by (see also (5.6.7))

q4α

bα = eαβγ aγ,β ,

eα = ϕ,α − a˙ α ,

(6.2.72)

where aα should satisfy a gauge condition; for the Lorentz gauge aα,α − µ0 ε0 ϕ˙ = 0 .

(6.2.73)

For small deformations all quantities referred to the intermediate state that appear in these perturbation equations can be replaced by the corresponding variables in the rigid-body state. To complete the description the form of the constitutive equations in this state will also be given. They are η0

− P0α

1 (m) 0 1 (m) 0 0 1 λ B − L B B − λ(e) E 0 ρ0 α α 2ρ0 αβ α β ρ0 α α

= c(Θ0 − Θ0 ) −

1 (e) 0 0 L E E , 2ρ0 αβ α β (e)

(e)

(em)

= −( 2 χαβ + 4 χαβγδ Eγ0 Eδ0 )Eβ0 − χβα Bβ0 (e)

(e)

−(λα + Lαβ Eβ0 )(Θ0 − Θ0 ) , (m)

(m)

(6.2.74) (em)

0 0 0 0 0 ML α = −( 2 χαβ + 4 χαβγδ Bγ Bδ )Bβ − χαβ Eβ (m)

−(λα J0α Q0α

(m)

+ Lαβ Bβ0 )(Θ0 − Θ0 ) ,

0 Θ,β , Θ0 0 0 = −κ0αβ Θ,β + βαβ Eβ0 , 0 0 = σαβ Eβ0 + ββα

6.2 Linearization of the Lorentz Model

169

0 0 where the coeﬃcients σαβ , βαβ and κ0αβ take the forms

Λ0αβ = Λαβ (0, B0γ , Θ0 ) (r)

(m)

(m)

(t)

= Λαβ + 3 Λαβγ Bγ0 + 4 Λαβγδ Bγ0 Bδ0 + Λαβ (Θ0 − Θ0 ) . (6.2.75) When these constitutive relations are substituted into the balance equations (6.2.38)–(6.2.40) and (6.2.42), the rigid-body problem is reduced to a problem for the 14 unknowns Eα0 , Bα0 , Q 0 , Θ0 , Ξα and Ωα . Again this problem could even further be reduced by introducing electromagnetic potentials according to Eα0 = Φ0,α − A˙ 0α , (6.2.76) Bα0 eαβγ A0γ,β , with A0α,α − µ0 ε0 Φ˙ 0 = 0 .

(6.2.77)

What remains, therefore, are now the boundary and jump conditions. 6.2.4 Decomposition of the Jump and Boundary Conditions The decomposition of the jump conditions is a very complex problem, in general, and this is the reason why we restricted ourselves already in Sect. 6.2.1 to singular surfaces of the second order. This restriction will be maintained here too. We shall be even more restrictive and shall assume simultaneously that there are no propagating singular surfaces in the intermediate state. ¯ N = 0. An acceleration wave, if present, can then only Hence, in this state W exist in the perturbed state; the speed of propagation in this perturbed state then also forms the total wave speed and may without confusion be called WN . The above special situation does not necessarily require, however, that WN be small. In fact, WN is a wave speed the numerical value of which follows from the material properties. In what follows it is advantageous to distinguish between the two diﬀerent kinds of singular surfaces, namely i) a propagating singular surface Σ (i) (acceleration wave) with velocity WN ; ii) a material surface Σ (ii) for which WN = 0. Since Σ (i) does not exist in the intermediate state, all quantities in this state must be continuous at Σ (i) , i.e. ¯ α = 0 , etc, at Σ (i) . E The linearization of the jump conditions is rather simple in a Lagrangean formulation, and from (6.2.9), (6.2.11) and (6.2.17) we immediately obtain

170

6 Linearization

bα Nα = 0 , eαβγ eβ Nγ + bα WN = 0 , a dα + pα Nα = 0 , eαβγ haβ − mβ Nγ − (daα + pα ) WN = 0 , jα Nα − q WN = 0 , (6.2.78) −1 ¯ ¯ β ( mβ Nγ − mγ Nβ ) tαβ Nβ = F¯γα Eγ pβ Nβ + B ¯ β pδ WN , +eβγδ B ¯ α bα − E ¯ α pα WN , qα Nα = ρ0 u + M all valid on Σ (i) . In the above, u denotes the perturbed internal energy, i.e. ˇ + η¯θ + Θs ¯ α pα − 1 P ¯ α eα . ¯ − 1E u = {ψˇ − (ψ)} ρ0 ρ0

(6.2.79)

¯ α, B ¯ α etc. need not be zero, but On the other hand, on Σ (ii) the jumps in E WN is. In this case the linearized jump conditions read bα Nα = 0 , eαβγ eβ Nγ = 0 , a dα + pα Nα = 0 , eαβγ haβ − mβ Nγ = 0 , jα Nα = 0 , −1 ¯ −1 ¯γ P ¯ β Nβ Fεα E tαβ Nβ = −F¯γδ ¯ β ( M ¯ β Nγ − M ¯ γ Nβ ) uδ,ε +B −1 ¯ γ pβ Nβ + P ¯ β Nβ eγ E +F¯γα ¯ β ( mβ Nγ − mγ Nβ ) + ( M ¯ β Nγ − M ¯ γ Nβ )bβ , +B qα Nα = 0 , (6.2.80) on Σ (ii) , whereas the jump conditions in the intermediate state follow from (6.2.9), (6.2.11) and (6.2.17) by invoking WN = 0 and writing all quantities with an overhead bar. This completes the linearization of the jump conditions. It should be noted that they can still be simpliﬁed (formally just a little bit, but for practical calculations enormously) when O(Eε)-terms are neglected, i.e. when the (¯·)ﬁelds are replaced by the rigid-body ﬁelds (·)0 . This step is a trivial one and will therefore be deleted here.

6.3 Linearisation of the Other Models and Comparison In the preceding section we presented the decomposition and linearization procedure for the Lorentz model. This would suﬃce, in general, because

6.3 Linearisation of the Other Models and Comparison

171

the proofs in Chaps. 4 and 5 showed that all formulations are equivalent anyhow. This is true, but since various formulations are used in the literature and because one should be able to compare these with our presentation, we shall in this section collect the balance laws and jump conditions for all ﬁve models. The comparison of the constitutive equations will be postponed until Sects. 6.4 and 6.5, however. To perform the comparison of the various models, recall that the results of Chap. 5 showed us that, in the Lagrangean formulation there are essentially only two diﬀerent formulations. The ﬁrst is the Chu-formulation (with its two diﬀerent stress models) the other one is the Lorentz model, which with a diﬀerent stress tensor also embraces the statistical model. The Maxwell–Minkowski model can be connected to either one of these two basic models. In the previous chapter the Lagrangean formulation of it was developed with the aid of the Chu-magnetization, rather than the Lorentzmagnetization, because most formulas took a much simpler form this way. In what follows we shall discuss therefore models (I,II,III) and (IV,V) as the two basic diﬀerent groups, although the separation of the various models could equally well be made according to (I,II) and (III,IV,V). We begin our comparison by stating the decoupled electromagnetic equations. They are obtained from equations (5.6.1)–(5.6.6) (compare also systems (6.2.21), (6.2.22) of the previous section). In the Chu formulation they read ¯C , ¯ a = −µ0 M B α,α α,α

¯˙ a + eαβγ E ¯˙ C , ¯ γ,β = −µ0 M B α α

¯a =Q ¯ −P ¯ α,α , D α,α

¯˙ a + eαβγ H ¯˙ α , ¯ γ,β = ¯ −D Jα + P α

(6.3.1)

¯˙ + ¯ Q Jα,α = 0 , and

˙a baα,α = −µ0 mC ˙C α,α , bα + eαβγ eγ,β = −µ0 m α , daα,α = q − pα,α ,

−d˙ aα + eαβγ hγ,β = jα + p˙ α ,

(6.3.2)

q˙ + jα,α = 0 , ¯ a, D ¯ a , ba and da are auxiliary ﬁelds, which, with the aid of (5.2.9) in which B α α α α and (6.2.23), become ¯ ¯ a = ε0 J¯C¯ −1 E D α αβ β ,

¯ a = µ0 J¯C¯ −1 H ¯ B α αβ β ,

(6.3.3)

and

−1 −1 ¯ −1 −1 ¯ −1 −1 ¯ −1 ¯ daα = ε0 J¯ C¯αβ Fγδ − C¯βγ Fαδ − C¯αγ Fβδ )Eβ uδ,γ , eβ + (C¯αβ −1 ¯ −1 ¯ baα = µ0 J¯ C¯ −1 hβ + (C¯ −1 F¯ −1 − C¯ −1 F¯ −1 − C¯αγ F )Hβ uδ,γ . αβ

αβ

γδ

βγ

αδ

βδ

(6.3.4)

172

6 Linearization

Note the symmetry of the formulas (6.3.4). Its origin is the complete symmetry of the Maxwell equations in this formulation. In the statistical and the Lorentz model the decomposition of the Maxwell equations was already made in Sect. 6.2.2. For completeness and for the purpose of comparison they will be repeated here. In the intermediate state they are ¯ α,α = 0 , B ¯˙ α + eαβγ E ¯ γ,β = 0 , B ¯ −P ¯ α,α , ¯a =Q D α,α

(6.3.5)

¯˙ a + eαβγ H ¯˙ α + eαβγ M ¯a =¯ ¯L , −D Jα + P α γ,β γ,β ¯˙ + ¯ Q Jα,α = 0 , whereas in the perturbed state they become bα,α = 0 , b˙ α + eαβγ eγ,β = 0 , daα,α = q − pα,α ,

(6.3.6)

−d˙ aα + eαβγ hγ,β = jα + p˙ α + eαβγ mL γ,β , q˙ + jα,α = 0 , ¯ a , da and ha as given in (6.2.25) and (6.2.26): ¯ a, H with D α α α α ¯ ¯ a = ε0 J¯C¯ −1 E D α αβ β , ¯ a = 1 C¯αβ B ¯ β − ε0 eβγδ C¯αβ F¯εγ ξ˙ε E ¯δ H α J¯ µ0 J¯

(6.3.7)

and −1 ¯ −1 ¯ ¯ C¯ −1 eβ + (C¯ −1 F¯ −1 − C¯ −1 F¯ −1 − C¯αγ Fβδ )Eβ uδ,γ } , daα = ε0 J{ αβ αβ γδ βγ αδ

1 ¯ 1 haα = ¯ Cαβ bβ − ε0 eβγδ F¯εγ ξ˙ε C¯αβ eδ J µ0 1 ¯ ¯ µ )(F¯δα δβγ + F¯δβ δαγ − C¯αβ F¯γδ + Bβ − eβνµ F¯εν ξ˙ε E µ0 ¯ δ C¯αβ F¯εγ u˙ ε . −ε0 eβγµ Eµ C¯αβ ξ˙δ uδ,γ − ε0 eβγδ E

(6.3.8)

Note that in the above sets of Maxwell equations we have retained the upper indices C and L for the magnetizations.

6.3 Linearisation of the Other Models and Comparison

173

Clearly, because polarization and magnetization are based on diﬀerent conceptions, we cannot expect the formulas (6.3.7) and (6.3.8) to be symmetric. This diﬀerence becomes formally particularly apparent if the equations in the intermediate state are approximated by those in the rigid-body state. Then the rigid-body motion does, formally, not enter the Chu-formulation, but it does show up in the Lorentz- formulation. We conclude this listing of the various forms of the Maxwell equations with the Maxwell–Minkowski formulation. The equations are ¯ α,α = 0 , B

¯˙ α + eαβγ E ¯ γ,β = 0 , B

¯ α,α = Q ¯ , D

¯˙ α + eαβγ H ¯ γ,β = ¯Jα , −D

(6.3.9)

¯˙ + ¯ Q Jα,α = 0 , bα,α = 0 ,

b˙ α + eαβγ eγ,β = 0 ,

dα,α = q ,

−d˙ α + eαβγ hγ,β = jα ,

(6.3.10)

q˙ + jα,α = 0 , and must be supplemented by the relations ¯ ¯ α = ε0 J¯C¯ −1 E ¯ D αβ β + Pα ,

¯ α = µ0 J¯C¯ −1 H ¯ ¯C B αβ β + µ0 Mα ,

(6.3.11)

and

−1 −1 ¯ −1 −1 ¯ −1 −1 ¯ −1 ¯ Fγδ − C¯βγ Fαδ − C¯αγ Fβδ )Eβ uδ,γ + pα , eβ + (C¯αβ dα = ε0 J¯ C¯αβ −1 −1 ¯ −1 −1 ¯ −1 −1 ¯ −1 ¯ Fγδ − C¯βγ Fαδ − C¯αγ Fβδ )Hβ uδ,γ + µ0 mC bα = µ0 J¯ C¯αβ eβ + (C¯αβ α.

(6.3.12) Apart from the use of diﬀerent auxiliary ﬁelds the most essential diﬀerence C in these formulations lies in the choice of the magnetization: ML α against Mα , which are related to each other according to (see (5.3.10)) ¯ C = J¯C¯ −1 M ¯L , M α β αβ ¯ ¯ −1 L ¯ ¯ −1 ¯ −1 ¯ −1 ¯ −1 ¯ −1 ¯ −1 ¯ L mC α = J Cαβ mβ + J(Cαβ Fγδ − Cβγ Fαδ − Cαγ Fβδ )Mβ uδ,γ .

(6.3.13)

Before we pass on to the mechanical balance laws, we would like to draw the reader’s attention to the basic diﬀerences in the Gauss and Faraday laws of the three formulations above. Because these laws are homogeneous in the Lorentz and in the Maxwell–Minkowski formulations electromagnetic potentials could easily and straightforwardly be introduced as was shown at the end of Sect. 6.2.4 (cf. (6.2.72), (6.2.73) or (5.6.7)). In the Chu formulation, on the other hand, these equations are inhomogeneous. Therefore, in this formulation the potentials Aα and Φ cannot be introduced in

174

6 Linearization

the same way as in Sect. 6.2.4. They must rather be deﬁned as ¯ = Φ¯ − A¯˙ , ¯a =e ¯C, A¯ − µ M E B α

αβγ

γ,β

0

α

baα = eαβγ aγ,β − µ0 mC α , with the gauge conditions ¯˙ = 0 , A¯α,α − ε0 µ0 Φ

α

,α

α

eα ϕ,α − a˙ α , aα,α − ε0 µ0 ϕ˙ = 0 .

(6.3.14)

(6.3.15)

The above listed electromagnetic equations must still be supplemented by two sets of constitutive relations for e.g. polarization and magnetization. They are all given in Chap. 5, and since they are directly derivable from a free energy functional the linearization is trivial and will, therefore, be omitted here. As was done for the Maxwell equations, we also wish to recapitulate the mechanical balance laws. If body forces of electromagnetic origin are not speciﬁed the equations can easily be taken over from (6.2.34) and (6.2.35). The momentum equations are ¨α = tαβ,β + ρ0 fαe , (6.3.16) ρ0 ξ¨α = δiα (T¯iβ,β + ρ0 F¯ie ) , ρ0 u whereas the energy equations in the intermediate and in the perturbed state can be written as ¯α − Q ¯ η¯˙ = ¯ ¯ α,α Jα E ρ0 Θ (6.3.17) ¯α + ¯ ¯ s = jα E ρ0 η¯˙ θ + ρ0 Θ˙ Jα eα − qα,α . All that is needed to complete the above equations for the various formulations is to prescribe the body force; for, all remaining quantities are either independent ﬁelds or else given by constitutive relations. The correspondence conditions of the various formulations of the latter were treated in Sect. 5.7 and will again be taken up in the following sections. Incidentally, that angular momentum is satisﬁed identically by satisfying objectivity requirements in the constitutive relations is reason for us not to list the balance law of moment of momentum. Moreover, it is noted that the reduced energy equation is the same in all formulations. Let us now list the body force expressions of the various formulations. In the Chu formulation we have, from (5.2.11)1 , for model I: −1 I ¯ −P ¯ β,β )Eα + eαβγ (¯ ¯˙ β )B ¯a (Q (ρ0 F¯ie ) = δiδ F¯αδ Jβ + P γ ¯C H ¯ ¯C¯ ¯ −1 ¯ ¯ , − µ0 M β,β α + Fβδ (Pα Eβ + µ0 Mα Hβ ) ,α (6.3.18) −1 ¯ I ¯ β + µ0 M ¯ C hβ + µ0 mC H ¯ β) (ρ0 fαe ) = II(ρ0 fαe ) + F¯βα (Pε eβ + pε E ε ε −1 ¯ −1 ¯ ¯ ¯ CH ¯ β )uγ,δ Fδα (Pε Eβ + µ0 M − F¯βγ , ε ,ε

where

II

(ρ0 fαe ) is listed in (6.3.19)2 . On the other hand,

6.3 Linearisation of the Other Models and Comparison

for model II: (from (5.2.12)1 ) −1 II ¯ α + eαβγ (¯ ¯˙ β )B ¯a ¯ −P ¯ β,β )E (ρ0 F¯ie ) = δiδ F¯αδ Jβ + P (Q γ ¯C H ¯ − µ0 M β,β α , −1 II ¯ β + (Q ¯ −P ¯ ε,ε )eβ + eβγδ (¯Jγ + P ¯˙ γ )ba (q − pε,ε )E (ρ0 fαe ) = F¯βα δ ¯ a − µ0 M ¯ C hβ − µ0 mC H ¯ +eβγδ (jγ + p˙ γ )B ε,ε ε,ε β δ ¯ −P ¯ ε,ε )E ¯ β + eβµν (¯ ¯˙ µ )B ¯a −F¯ −1 F¯ −1 (Q Jµ + P βγ

δα

175

(6.3.19)

ν

¯CH ¯ −µ0 M ε,ε β uγ,δ . In actual calculations one only needs the force expression for one single model, because the stress tensors of the two models are related by (5.2.22), or I

T¯iα =

I¯ tαβ

=

II

−1 ¯ ¯ ¯ CH ¯ β) , T¯iα − δiα F¯βδ (Pα Eβ + µ0 M α

II ¯ tαβ

C ¯ β eγ + µ0 mC H ¯γ + P ¯ ¯C − F¯γα (pβ E β γ + µ0 Mβ hγ )

(6.3.20)

−1 ¯ −1 ¯ ¯ ¯ CH ¯ ε )uγ,δ . Fδα (Pβ Eε + µ0 M +F¯εγ β

When the stress tensor and the body force of model I are substituted into the momentum equation what results is the body force of model II and a term which agrees with II Tiα . In the Maxwell–Minkowski model III the body force is given by (5.4.5)1 . When decomposed this becomes −1 ¯ III ¯β E ¯γ + P ¯ β,α + µ0 M ¯ CH ¯ β,α (ρ0 F¯ie ) = δiδ F¯αδ Jβ B QEα + eαβγ ¯ β ¯βB ¯˙ β B ¯αE ¯˙ γ + D ¯ γ ) + F¯ −1 (P ¯ β + µ0 M ¯ CH ¯ β) , + eαβγ (D α βδ,α −1 III ¯ β + Qe ¯ β + eβγδ (¯ ¯ γ eγ,β ¯δ) + P qE (ρ0 fαe ) = F¯βα Jγ bδ + jγ B ¯ ˙ ¯˙ ¯ γ,β + µ0 M ¯ C hγ,β + µ0 mC H ¯ +pγ E γ γ γ,β + eβγδ (Dγ bδ + Dγ bδ ¯˙ δ + d˙ γ B ¯ δ ) + F¯ −1 P ¯β ¯ γ eβ + pγ E +dγ B βα,γ

¯ C hβ + µ0 mC H ¯ β − F¯ −1 F¯ −1 QE ¯ν ¯ β + eβµν ¯Jµ B +µ0 M γ γ βγ δα ¯εE ¯ µB ¯˙ µ B ¯˙ ν + D ¯ ε,β + µ0 M ¯ CH ¯ ε,β + eβµν (D ¯ ν ) uγ,δ +P ε −1 ¯ −1 ¯εE ¯ β + µ0 M ¯ CH ¯ β) . Fδα uγ,δ ),ε (P −(F¯βγ ε

(6.3.21)

176

6 Linearization

We recall that the stress tensors in models I and III are identical, hence I

T¯iα =

III

T¯iα

and

I

tαβ =

III

tαβ .

(6.3.22)

The relationships between ( II T¯iα , II tαβ ) and ( III T¯iα , III tαβ ) can then easily be read oﬀ from (6.3.20). Next, we list the body forces for the statistical and the Lorentzformulations. In the Lorentz-formulation (model V) they are already given in (6.2.36) and (6.2.37). For reasons of comparison they will be repeated here: −1 V ¯˙ β )B ¯γ ¯ −P ¯ β,β )Eα + eαβγ (¯ (ρ0 F¯ie ) = δiδ F¯αδ Jβ + P (Q ¯L −M ¯ L )B ¯β , + (M α,β β,α −1 V ¯ −P ¯ ε,ε )eβ + (q − pε,ε )E ¯ β + eβγδ (¯Jγ + P ¯˙ γ )bδ (Q (ρ0 fαe ) = F¯βα (6.3.23) ¯δ + B ¯ γ (mL − mL ) +eβγδ (jγ + p˙ γ )B β,γ γ,β ¯ −P ¯ ε,ε )E ¯L −M ¯ L ) − F¯ −1 F¯ −1 (Q ¯β + bγ (M β,γ γ,β βγ δα ¯˙ µ )B ¯ν + B ¯ ε (M ¯L −M ¯ L ) uγ,δ . +eβµν (Jµ + P β,ε ε,β On the other hand, in model IV we have (cf. (5.3.19)1 ) IV ¯ αP ¯ LB ¯β − M ¯ ¯ L¯ (ρ0 F¯ e ) = V(ρ0 F¯ e ) + δiδ F¯ −1 (E α β + δαβ Mγ Bγ ) i

i

αδ

−1 ¯ IV ¯ L bε ¯ε − M (ρ0 fαe ) = V(ρ0 fαe ) + F¯βα Eβ pε + eβ P β L¯ ¯L ¯ −mL β Bε + δβε (Mγ bγ + mγ Bγ ) −1 ¯ −1 ¯ ¯ ¯ LB ¯ ε + δβε M ¯ LB ¯ µ uγ,δ Fδα Eβ Pε − M − F¯βγ µ β

,ε

,β

,

(6.3.24)

.

These forces need not be calculated, however, because the divergence term on the right-hand side of (6.3.24) will, with opposite sign, also arise in the stress tensor of the statistical formulation; hence, the momentum equation remains unchanged. Indeed, (cf. (5.3.22)) −1 ¯ ¯ ¯ αM ¯ L + δαβ B ¯γM ¯ L) , T¯iα − δiδ F¯βδ (Pα Eβ − B γ β −1 ¯ β eγ − bβ M ¯ β mL ¯γ + P ¯L−B pβ E = V tαβ − F¯γα γ γ ¯ δ mL ) ¯L+B +δβγ (bδ M δ δ −1 ¯ −1 ¯ ¯ ¯βM ¯ L + δβε B ¯ µM ¯ L uγ,δ . Fδα Pβ Eε − B +F¯εγ ε µ

IV

T¯iα =

IV

tαβ

V

(6.3.25)

To ﬁnd the link between the groups (I, II, III) and (IV, V ) we also need the decomposed versions of relation (5.6.12) between V Tiα and II Tiα . This relation gives

6.3 Linearisation of the Other Models and Comparison

177

¯ αH ¯ aH ¯γH ¯β − B ¯ a − 1 δαβ (B ¯γ − B ¯ a) , ¯ aH B α γ γ β 2

−1 T¯iα + δiδ F¯βδ V −1 ¯ a hγ − bβ H ¯ β ha ¯γ + B ¯a−B tαβ = II tαβ + F¯γα baβ H γ γ β ¯ a hδ − bδ H ¯ δ ha ) ¯δ + B ¯a−B − 12 δβγ (baδ H δ δ δ −1 ¯ −1 ¯ a ¯ ¯βH ¯ aH ¯ µH ¯ a − 1 δβε (B ¯µ − B ¯ a ) uγ,δ . Fδα Bβ Hε − B −F¯εγ ε µ µ 2 (6.3.26) With the aid of the formulas (6.2.25)2 , (6.2.26)2 , (6.3.3)2 and (6.3.4)2 the ¯ a , ba and ha can be eliminated. ¯ a, H auxiliary ﬁelds B α α α α If we wished to do so, we could also give the decoupled constitutive equations for the stresses I Tiα , . . . , V Tiα . We shall not do it here and restrict ourselves to recalling that only II Tiα and V Tiα are directly derivable from a free energy; hence, their linearization is trivial. However, the free energies occurring in these relations are not identical; they are related according to (5.6.20), which when decoupled yields V

T¯iα =

II

µ0 ¯ CM ¯C, − ¯C¯αβ M α β 2J V ˇ − V(ρ0 ψ) ˇ = II(ρ0 ψ) ¯ − II(ρ0 ψ) ¯ − µ0 C¯αβ M ¯ C mC (ρ0 ψ) α β J¯ µ0 ¯ C ¯ C ¯ −1 ¯ + ¯M M Fγδ Cαβ − δαδ F¯γβ − δβδ F¯γα uγ,δ , J α β V(ρ ψ) ˇ 0

=

II(ρ ψ) ¯ 0

(6.3.27)

We conclude this section with a survey of the decomposed jump and boundary conditions. As before, the singular surfaces will be assumed to be of order 2. Since the deduction of the respective conditions is straightforward and analogous to the methods illustrated in Sect. 6.2.4, we shall only present the ﬁnal results. In accord with the assumptions laid down in Sect. 6.2.4, Σ (i) will denote the propagating surface of acceleration waves across which the intermediate ﬁelds (¯·) do not suﬀer a jump ( (¯·) = 0, on Σ (i) ). On a material surface Σ (ii) , on the other hand, WN = 0, but the intermediate ﬁelds may jump there. We start with the jump conditions for the electromagnetic ﬁelds. One obtains: for model I and model II a ¯ C Nα = 0 ,  ¯ + µ0 M B α α     a   ¯ α Nα = 0 , ¯ +P D  α    ¯ β Nγ = 0 , eαβγ E     ¯ β Nγ = 0 ,  eαβγ H      ¯ Jα Nα = 0 ,

on Σ (ii) ,

(6.3.28)

178

6 Linearization

and  baα + µ0 mC α Nα = 0 ,     a   dα + pα Nα = 0 ,     a C on Σ (i) and Σ (ii) , (6.3.29) eαβγ eβ Nγ + (bα + µ0 mα ) WN = 0 ,     eαβγ hβ Nγ − (daα + pα ) WN = 0 ,       jα Nα − q WN = 0 , for model III # ¯ α Nα = 0 , ¯ α Nα = 0 , B D on Σ (ii) , (6.3.30) ¯ β Nγ = 0, eαβγ H ¯ β Nγ = 0, ¯ eαβγ E Jα Nα = 0 , and  bα Nα = 0 , dα Nα = 0 ,       eαβγ eβ Nγ + bα WN = 0 ,  on Σ (i) and Σ (ii) ,   eαβγ hβ Nγ − dα WN = 0 ,      jα Nα − q WN = 0 ;

for model IV and model V ¯ α Nα = 0 , B

          

¯ α Nα = 0 , ¯α + P D ¯ β Nγ = 0 , eαβγ E on Σ (ii) ,    ¯ L Nγ = 0 ,  ¯β − M  eαβγ H  β     ¯ Jα Nα = 0 ,

(6.3.31)

and a  bα Nα = 0 , dα + pα Nα = 0 ,        eαβγ eβ Nγ + bα WN = 0 , on Σ (i) and Σ (ii) . a  a  N N eαβγ hβ − mL − d + p = 0 , γ α α  α β     jα Nα − q WN = 0 , These equations can easily be speciﬁed for Σ (i) and Σ (ii) .

(6.3.32)

(6.3.33)

6.3 Linearisation of the Other Models and Comparison

179

For the jump conditions of momentum and energy of matter and ﬁelds we must distinguish between Σ (i) and Σ (ii) . We ﬁrst list those valid at the wave surface Σ (i) ; they only need be given for the perturbed relations. One obtains: for model I and model III I tαβ Nβ = III tαβ Nβ −1 ¯ C hγ Nβ ¯ β eγ Nβ − µ0 M = F¯γα −P β ¯ a pδ WN , +eβγδ B β ¯ α µ0 mC WN ¯ α pα − H qα Nα = ρ0 II u − E α

          on Σ         .

(i)

. (6.3.34)

We note that in the ﬁrst of the above conditions the symmetry between the electric and the magnetic ﬁelds, which is characteristic for the Chuformulation, is destroyed, because in the derivation the term ¯ C WN eβγδ Dβa µ0 M δ was dropped as it is proportional to c−2 . Furthermore, for model III we have ¯ a by replaced in (6.3.34)1 B β ¯ β − µ0 M ¯C . B β For model II we have II −1 ¯ ¯ γ µ0 mC Nβ tαβ Nβ = F¯γα Eγ pβ Nβ + H β ¯ a pδ WN , +eβγδ B β ¯ α µ0 mC WN ¯ α pα − H qα Nα = ρ0 II u − E α Moreover, for model IV IV −1 ¯ L bβ Nγ ¯ β e γ Nβ − M tαβ Nβ = F¯γα −P β ¯ β pδ WN , +eβγδ B ¯ α pα WN , ¯ α bα − E qα Nα = ρ0 V u + M and for model V V −1 ¯ ¯ β ( mL Nγ tαβ Nβ = F¯γα Eγ pβ Nβ + B β

¯ , − mL γ Nβ ) + eβγδ Bβ pδ WN ¯ α pα WN , ¯ L bα − E qα Nα = ρ0 V u + M α

       on Σ (i) .      ,        on Σ (i) ,                   

on Σ (i) .

(6.3.35)

(6.3.36)

(6.3.37)

180

6 Linearization

Since in all of the above jump conditions for the energy, qα is the same function (needless to say once more that in model V qα stands for qSα rather than qL α ) the right-hand sides of these conditions must be identical also. That this is indeed the case can most easily be seen from (5.6.19), which in perturbed form reads ¯ C − µ0 H ¯ α mC − µ0 C¯αβ M ¯ C mC ρ0 V u = ρ0 II u − µ0 hα M α α α β J¯ µ0 ¯ C ¯ C ¯ −1 ¯ + ¯M M F Cαβ − δαδ F¯γβ − δβδ F¯γα uγ,δ . 2J α β γδ

(6.3.38)

Finally, we list the boundary conditions for momentum and energy on material surfaces Σ (ii) . On these, WN vanishes but the intermediate ﬁelds may jump instead. Consequently , for model I and model III I T¯iα Nα = III T¯iα Nα −1 ¯β E ¯ C H ¯ α Nβ − µ0 M ¯ α Nβ , P = δiδ F¯αδ β I tαβ Nβ = III tαβ Nβ −1 ¯ β eγ Nβ − pβ E ¯ γ Nβ −P = F¯γα ¯ C hγ Nβ − µ0 mC H ¯ γ Nβ −µ0 M β β −1 ¯ −1 ¯β E ¯ C H ¯ γ Nβ + µ0 M ¯ γ Nβ uδ,ε Fεα P +F¯γδ β

                

,

               

(6.3.39)

¯ a and ba must, respectively, be replaced on Σ (ii) , in which for model III, B α α by ¯ α − µ0 M ¯ C) and (bα − µ0 mC (B α α) . For model II II −1 ¯ α P ¯ α µ0 M ¯ C Nβ , ¯ β Nβ + H E T¯iα Nα = δiδ F¯αδ β II −1 ¯ γ pβ Nβ + eγ P ¯ β Nβ tαβ Nβ = F¯γα E ¯ γ µ0 mC Nβ + hγ µ0 M ¯ C Nβ +H β β −1 ¯ −1 ¯γ P ¯ γ µ0 M ¯ C Nβ uδ,ε ¯ β Nβ + H Fεα E −F¯γδ β on Σ (ii) ,

                  ,

(6.3.40)

6.4 Interchanging Dependent and Independent Constitutive Variables

181

for model IV −1 ¯β E ¯ L B ¯ α Nβ − M ¯ β Nα , −P T¯iα Nα = δiδ F¯αδ β IV −1 ¯ β eγ Nβ − pβ E ¯ γ Nβ −P tαβ Nβ = F¯γα ¯ L bβ Nγ − mL B ¯ β Nγ −M β β −1 ¯ −1 ¯ L B ¯β E ¯ γ Nβ + M ¯ β Nγ uδ,ε Fεα P +F¯γδ β

IV

                  ,

(6.3.41)

on Σ (ii) , and for model V V

−1 ¯ α P ¯ β Nβ E T¯iα Nα = δiδ F¯αδ

L L ¯ β ( M ¯ Nα − M ¯ Nβ ) , +B α β V −1 ¯ γ pβ Nβ + eγ P ¯ β Nβ E tαβ Nβ = F¯γα ¯ β ( mL Nγ − mL Nβ ) +B γ β L L ¯ Nγ − M ¯ Nβ ) +bβ ( M γ β −1 ¯ −1 ¯γ P ¯ β Nβ Fεα E −F¯γδ L L ¯ β ( M ¯ Nγ − M ¯ Nβ ) uδ,ε +B γ β

                                          ,

(6.3.42)

on Σ (ii) . The boundary conditions of energy on Σ (ii) are equal in all formulations and simply read ¯ α Nα = qα Nα = 0 , Q on Σ (ii) . (6.3.43) Needless to say here that, with the use of relations (6.3.20), (6.3.25), (6.3.26) and (6.3.38), all these jump conditions and boundary conditions can be transformed into each other.

6.4 The Meaning of Interchanging Dependent and Independent Constitutive Variables in one Formulation In Chaps. 4 and 5 it was demonstrated that all theories of magnetizable and polarizable thermoelastic materials are fully equivalent. The equivalence requirements could be stated as interrelationships between the free energies and heat ﬂux vectors of the formulations being compared. It was further

182

6 Linearization

demonstrated in Sect. 5.7 of Chap. 5 that full equivalence of diﬀerent formulations can be destroyed simply by comparing energy expressions, which are too restrictive to allow complete matching. In particular, using in each theory a polynomial expansion of the free energy in terms of its variables and truncating these expansions at a certain prescribed level, destroyed complete agreement. One can, of course, insist in polynomial representations, but must then accept the fact that equivalence of two formulations is only possible to within terms not being omitted in the expansion process. A similar situation prevails if one tries to compare theories, which are based on one single model (say the Lorentz model), in which diﬀerent dependent and independent constitutive variables are used. The situation is similar as in the nonlinear theory of elasticity, in which there also exist two diﬀerent constitutive formulations, one in which the free energy is a function of the strains and a second one, whose energy function is obtained from the former through a Legendre transformation. This latter energy, which is called enthalpy (or complementary energy) is a function of the stresses rather than the strains. It gives the strains as functions of the stresses. Inverting the stress-strain relationship, that is expressing stress as a function of strain then eﬀectively amounts to ﬁnding the free energy from the enthalpy. If these inversions can be performed, the theories based on enthalpy and free energy, respectively, are fully equivalent. It is known that this step is not a trivial one, except in the linear elasticity theory, in which it amounts to writing the elasticity coeﬃcients in terms of the compliances. In an electromechanical interaction theory changes of dependent and independent variables are possible also among the electromagnetic constitutive variables. Indeed, it is not the change of strain and stress as dependent and independent constitutive variables, that ordinarily gives rise to diﬀerent constitutive approaches, but rather that of electromagnetic variables. We showed in Chap. 3 already that there are four diﬀerent constitutive theories in just one formulation, not counting that stress and strain and temperature and entropy could also be interchanged as dependent and independent variables. In what follows we shall only be dealing with two possibilities, namely the Lorentz formulation, in which (Eα , Bα ) and (Pα /ρ0 , Mα /ρ0 ) are taken as the respective independent constitutive variables. Results for the Chu-formulation and other variable combinations will be stated, however. In the Lorentz model and if Eα and Bα are selected we have 1 ˇ αβ , Eα , Bα , Θ) , ψˇ = U − ηΘ − Eα Pα = ψ(E ρ0 and η=−

∂ ψˇ , ∂Θ

ML α = −ρ0

Pα = −ρ0

∂ ψˇ , ∂Bα

V

∂ ψˇ , ∂Eα

P Tαβ = ρ0

∂ ψˇ . ∂Eαβ

(6.4.1)

(6.4.2)

6.4 Interchanging Dependent and Independent Constitutive Variables

183

On the other hand, when Pα /ρ0 and ML α /ρ0 are the independent ﬁelds, we use as energy functional Pα ML 1 α ˆ B = ψ E , , , Θ , (6.4.3) ψˆ = U − ηΘ + ML α αβ ρ0 α ρ0 ρ0 and from it we obtain ∂ ψˆ , ∂Θ ∂ ψˆ Bα = , ∂ML α /ρ0

η=−

∂ ψˆ , ∂Pα /ρ0 ∂ ψˆ V P Tαβ = ρ0 . ∂Eαβ

Eα =

(6.4.4)

ˆ (6.4.1) and (6.4.3), the two constitutive In view of the deﬁnitions of ψˇ and ψ, theories lead to identical results if 1 1 ψˆ = ψˇ + Eα Pα + Bα ML α , ρ0 ρ0 or with the use of (6.4.2), if Pα ML α ˆ , ,Θ ψ Eαβ , ρ0 ρ0 ˆ ˆ ∂ ψ ∂ ψ ∂ ψˆ Pα ∂ ψˆ ML α = ψˇ Eαβ , , ,Θ + + . L ∂Pα /ρ0 ∂Mα /ρ0 ∂Pα /ρ0 ρ0 ∂ML /ρ ρ 0 0 α

(6.4.5)

(6.4.6)

Of course, there is also a dual relation to (6.4.6), namely ∂ ψˇ ∂ ψˇ ∂ ψˇ ∂ ψˇ ˆ ˇ ,− , Θ + Eα + Bα . ψ(Eαβ , Eα , Bα , Θ) = ψ Eαβ , − ∂Eα ∂Bα ∂Eα ∂Bα (6.4.7) ˆ ˇ For given functionals ψ and ψ, equations (6.4.6) and (6.4.7) must be satisﬁed identically if the two constitutive theories are to be equivalent. If only one of ˇ is given, then (6.4.6) or (6.4.7) are functional the energy functionals, ψˆ or ψ, diﬀerential equations to determine the other. The solution to these remains an open problem, as we shall not attack it here. Nevertheless, in order to demonstrate that this problem is extremely complex in general, consider, as a more or less arbitrary example, the following energy functional ˆ αβ , Mα ) = f(M2 ) + 1 λ(Eαα )2 + 2GEαβ Eαβ , (6.4.8) ψˆ = ψ(E 2ρ0 which may be regarded as the most simple functional form for an isotropic nonlinearly magnetizable body. In (6.4.8) M2 :=

1 L L M M , ρ20 α α

(6.4.9)

184

6 Linearization

and f(M2 ) is a continuous, diﬀerentiable function. With the aid of (6.4.4) we may derive the following expression for Bα Bα = 2

df ML α . dM2 ρ0

(6.4.10)

ˇ Substitution of this into the identity (6.4.5) allows determination of ψ, ˇ αβ , Bα ) = f(M2 ) − 2M2 df + 1 {λ(Eαα )2 + 2GEαβ Eαβ } . (6.4.11) ψ(E dM2 2ρ0 The right-hand side of (6.4.11) can be determined as an explicit function of Bα , or more precisely of B2 = Bα Bα , only if relation (6.4.10) is invertible, i.e., only if M2 is expressible as a function of Bα . This is rarely the case in general. However, the special choice M 1/2 4χ 1+2 ξ dξ 2χ 2

f(M2 ) = (∼ =

1 2 2χ

0

1 2 2 2 χM

+

1 4 4 4 χM

(6.4.12)

+ ···) ,

which is a nonlinear representation that may be regarded as an extension of the usual functional dependencies of the free energy on magnetization, gives (for 2 χ = 0, 4 χ = 0) M2 =

8 4 χ 2 1/2 , −1 + 1 + B 3 4 4χ 2χ 2χ

(6.4.13)

so that ψˇ may be written as 2 1/2 1/2 M 4χ 4χ ψˇ = dξ − 2 χ 1 + 2 M2 M2 1+2 ξ 2χ 2χ 0 1 λ(Eαα )2 + 2GEαβ Eαβ , + 2ρ0

1 2 2χ

(6.4.14)

in which M2 is given by (6.4.13). Often the construction of a free energy of one formulation from that of another is a matter of shear patience or simply becomes impossible analytically. This does not mean that equivalence is not possible in these cases; it simply means that a free energy function of the second formulation is given only implicitly. Such is already the case in the above representation if we leave f(M2 ) unspeciﬁed. The above construction of a free energy may appear to be rather artiﬁcial, because the representation (6.4.8) is of very limited practical applicability. Nevertheless, it is important from a mathematical point of view, because it explicitly demonstrates that the functional diﬀerential equations mentioned

6.4 Interchanging Dependent and Independent Constitutive Variables

185

above do indeed admit exact solutions, at least for the demonstrated case. Conditions imposed on the free energy functions that guarantee the existence of such solutions would be of value, and in particular, it would be valuable if, for instance, free energies could be constructed, which would still be of physical relevance, but would not admit a solution of the functional diﬀerential equations. In that case non-equivalence of two theories would be demonstrated. We shall not go any deeper into this subject, but will mention one simple and physically important case, in which existence of solutions of equations (6.4.6) or (6.4.7) can easily be established and for which equivalence of the theories compared is guaranteed. What we have in mind is the case for which the free energy functions can be expressed as Taylor series expansions about zero deformation, constant temperature and zero electromagnetic ﬁelds. The proof to this case will not be outlined here, but from the procedure explained below the reader should be able to construct his own proof. With these few comments we shall now leave the subject of an exact determination of the functional ψˇ from ψˆ and pass on to an approximate satisfaction of relations (6.4.6) and (6.4.7). To this end, we shall choose truncated polynomials as expressions for the free energies. If the degree of the polynomial representation of ψˆ is known, then with (6.4.6) that of ψˇ can be determined. In general the order of truncation of ψˇ needed to obtain full ˆ We shall not be so general and equivalence is not the same as that for ψ. choose for both energy functionals the same polynomial expressions. In complexity, we shall be as general as we were in (6.2.46) and thus write 1 1 (em) 1 (m) (m) ˇαβ Bα Bβ + ˇαβγδ Bα Bβ Bγ Bδ + χ ˇ Bα Eβ ψˇ = 2χ 4χ 2ρ0 4ρ0 ρ0 αβ +

1 1 1 (e) (e) ˇ Eα Eβ + ˇ Eα Eβ Eγ Eδ − ˇc(Θ − Θ0 )2 2χ 4χ 2ρ0 αβ 4ρ0 αβγδ 2

+

1 ˇ (m) 1 ˇ (m) L Bα Bβ (Θ − Θ0 ) λ Bα (Θ − Θ0 ) + ρ0 α 2ρ0 αβ

1 ˇ (e) 1 ˇ (e) L Eα Eβ (Θ − Θ0 ) Eα (Θ − Θ0 ) + + λ ρ0 α 2ρ0 αβ 1 1 ˇ(m) 1 (e) (m) + εˇαβγ Bβ + bαβγδ Bα Bβ + εˇβγδ Eβ ρ0 2ρ0 ρ0 1 ˇ(e) 1 + Eα Eβ − νˇγδ (Θ − Θ0 ) Eγδ + ˇcαβγδ Eαβ Eγδ , b 2ρ0 αβγδ 2ρ0 and

(6.4.15)

186

6 Linearization L L ML ML ρ30 (m) ML ρ0 (m) ML β β Mγ Mδ α α χ ˆ + χ ˆ ψˆ = 2 αβ 4 αβγδ 2 ρ0 ρ0 4 ρ0 ρ0 ρ0 ρ0 L M P (em) α β +ρ0 χ ˆαβ ρ0 ρ0 ρ0 (e) Pα Pβ ρ3 (e) Pα Pβ Pγ Pδ + 2χ ˆαβ + 0 4χ ˆαβγδ 2 ρ0 ρ0 4 ρ0 ρ0 ρ0 ρ0 L M (m) α ˆα − 12 ˆc(Θ − Θ0 )2 + λ (Θ − Θ0 ) ρ0 ML ρ0 ˆ (m) ML β α ˆ (e) Pα (Θ − Θ0 ) + L (Θ − Θ0 ) + λ α αβ 2 ρ0 ρ0 ρ0 L ML ρ0 ˆ (e) Pα Pβ ρ0 ˆ(m) ML β (m) Mβ α b + L (Θ − Θ ) + ε ˆ + 0 βγδ 2 αβ ρ0 ρ0 ρ0 2 αβγδ ρ0 ρ0 ρ0 ˆ(e) Pα Pβ (e) Pβ +ˆ εβγδ + b − νˆγδ (Θ − Θ0 ) Eγδ αβγδ ρ0 2 ρ0 ρ0 1 + ˆcαβγδ Eαβ Eγδ . 2ρ0

(6.4.16)

With these representations full equivalence is not possible, because the transformations indicated by (6.4.6) and (6.4.7) lead to terms, which have been omitted in the formulation of the respective energy functions. Nevertheless, except for these terms equivalence can be established. If (6.4.15) and (6.4.16) are used to exploit (6.4.6) and (6.4.7) in this approximate sense a series of identities can be derived for the phenomenological coeﬃcients of the two formulations. We have done this; the calculations for the derivation of the corresponding relations are very long. Unfortunately, the emerging identities are much too long, and conclusions that can be drawn from them in this full generality are very meagre in order to justify to list them here. Nevertheless, one result derivable from these identities may be quoted. It reads: If the co(m) (e) (em) ˆαβ , 2 χ ˆαβ , χ ˆαβ vanish, all remaining coeﬃcients accounting for eﬃcients 2 χ electromagnetic eﬀects must also vanish if the theories are to be equivalent in ˆ(m) , 2 χ ˆ(e) and χ ˆ(em) vanthe above mentioned sense. Otherwise stated, if 2 χ ish in one theory and the two theories are to be equivalent, the free energies reduce to ψ=

1 cαβγδ Eαβ Eγδ − ναβ (Θ − Θ0 )Eαβ − 12 c(Θ − Θ0 )2 , 2ρ0

and no other terms. In a truly polarizable and magnetizable material at least (m) (e) (em) one of the coeﬃcients 2 χαβ , 2 χαβ or χαβ must therefore be non-zero. In the following we shall exploit the equations (6.4.6) and (6.4.7) for an isotropic body, in which

6.4 Interchanging Dependent and Independent Constitutive Variables (m)

λα

(e)

= λα = 0 ,

(m) 2 χαβ

= 2 χ(m) δαβ ,

(m)

(e)

εαβγ εαβγ = 0 , (e) 2 χαβ

= 2 χ(e) δαβ ,

(m) 4 χαβγδ

= 4 χ(m) 13 (δαβ δγδ + δαγ δβδ + δαδ δβγ ) ,

(e) 4 χαβγδ

= 4 χ(e) 31 (δαβ δγδ + δαγ δβδ + δαδ δβγ ) ,

(m)

Lαβ = L(m) δαβ , (m)

(m)

(e)

(e)

187

(e)

Lαβ = L(e) δαβ ,

(em)

χαβ = 0 ,

(6.4.17)

ναβ = νδαβ ,

(m)

bαβγδ = b1 δαβ δγδ + 12 b2 (δαγ δβδ + δαδ δβγ ) , (e)

bαβγδ = b1 δαβ δγδ + 12 b2 (δαγ δβδ + δαδ δβγ ) , cαβγδ = λδαβ δγδ + G(δαγ δβδ + δαδ δβγ ) . Substituting these expressions into (6.4.15) and (6.4.16), we obtain 1 1 1 ψˇ = ˇ(m) Bα Bα + ˇ(m) (Bα Bα )2 + ˇ(e) Eα Eα 2χ 4χ 2χ 2ρ0 4ρ0 2ρ0 1 + ˇ(e) (Eα Eα )2 − 12 ˇc(Θ − Θ0 )2 4χ 4ρ0 1 ˇ (m) 1 ˇ (e) L Bα Bα (Θ − Θ0 ) + L Eα Eα (Θ − Θ0 ) + 2ρ0 2ρ0 1 ˇ(m) ˇ(e) Eα Eα )Eββ (b Bα Bα + b + 1 2ρ0 1 1 ˇ(m) ˇ(e) Eα Eβ )Eαβ − νˇ(Θ − Θ0 )Eαα + (b Bα Bβ + b 2 2ρ0 2 1 ˇ ˇ αβ Eαβ . + λ(Eαα )2 + 2GE 2ρ0

(6.4.18)

L L 2 ρ0 (m) ML ρ30 (m) ML ρ0 (e) Pα Pα α Mα α Mα ˆ ψ= ˆ + ˆ + ˆ 2χ 4χ 2χ 2 ρ0 ρ0 4 ρ0 ρ0 2 ρ0 ρ0 2 3 Pα Pα ρ + 0 4χ ˆ(e) − 12 ˆc(Θ − Θ0 )2 4 ρ0 ρ0 L ρ0 ˆ (m) ML ρ0 ˆ (e) Pα Pα α Mα + L (Θ − Θ0 ) + L (Θ − Θ0 ) 2 ρ0 ρ0 2 ρ0 ρ0 L ρ0 ˆ(m) ML α Mα ˆ(e) Pα Pα Eββ + +b b1 1 2 ρ0 ρ0 ρ0 ρ0 L ML P P ρ0 ˆ(m) Mα β α β (e) ˆ b2 + +b Eαβ 2 2 ρ0 ρ0 ρ0 ρ0 1 ˆ ˆ αβ Eαβ . −ˆ ν (Θ − Θ0 )Eαα + λ(Eαα )2 + 2GE 2ρ0

(6.4.19)

and

188

6 Linearization

With these exploitation of (6.4.6) stays within reasonable eﬀort. The approach is analogous to that demonstrated in Sect. 5.7 and at the beginning of this section. One simply evaluates Eα and Bα according to (6.4.4), L L ˆ (m) ML (Θ − Θ0 ) ˆ(m) ML ˆ(m) ML Bα = 2 χ α + 4χ α β Mβ Mα + L (m)

(m)

ˆ ML Eββ + b ˆ ML Eαβ , +b α 1 2 β ˆ (e) Pα (Θ − Θ0 ) ˆ(e) Pα + 4 χ ˆ(e) Pβ Pβ Pα + L Eα = 2 χ (e)

(6.4.20)

(e)

ˆ Pα Eββ + b ˆ Pβ Eαβ , +b 1 2 expresses with their use ψˇ as a function Pα /ρ0 and ML α /ρ0 rather than Eα and Bα and substitutes the emerging relation together with (6.4.16) into (6.4.6). When this is done, the following identities are obtained: 1 µ0 µ µ = = (m) , (m) µ − µ χ ˇ χ 0 2 4 µ ˆ(m) = ˇ(m) , 4χ 4χ χ(m) 2 µ (m) ˇ (m) , ˆ L L = χ(m) 2 µ ˆ(m) = ˇ(m) , b b 1,2 1,2 χ(m) cw , νˆ = νˇ = ν , ˆc = ˇc = Θ0 ˆ(m) 2χ

=

1 1 = − (e) = (e) , χ ˇ χ 2 4 1 (e) ˆ = ˇ(e) , 4χ 4χ χ(e) 2 1 (e) ˇ (m) , ˆ L L = χ(e) 2 1 ˆ(e) = ˇ(e) , b b 1,2 1,2 χ(e) ˆ=λ ˇ=λ, ˆ=G ˇ=G, λ G ˆ(e) 2χ

(6.4.21) with obvious inversions, which we shall not write down. In the above relations, µ is called magnetic permeability, χ(m) magnetic susceptibility and χ(e) electric susceptibility; for an ideal medium (i.e. a rigid, isotropic body) they are deﬁned by B = µH ,

M = χ(m) H ,

P = χ(e) H .

Further, cW is the speciﬁc heat, ν the thermoelastic constant, and λ and G ´ constants. In an isotropic body, therefore, the classical phenomare the Lame enological coeﬃcients are equal and thus clearly deﬁned. This is not so for all other coeﬃcients, as can be seen from (6.4.21). In fact, the transformations all involve the magnetic and electric susceptibility. The deﬁnitions of fourthorder electromagnetic, magnetostrictive, electrostrictive, thermoelectric and thermomagnetic coeﬃcients are not unique in this restricted isotropic theory. Care should therefore be observed with the use of speciﬁc names for these eﬀects. Note also that the above described transformation becomes simply impossible whenever 2 χ(m) and/or 2 χ(e) are zero. In this case, the outlined procedure does not lead to a polynomial expression for ψˆ of the form (6.4.19).

6.4 Interchanging Dependent and Independent Constitutive Variables

189

In the above only two possible constitutive formulations were investigated, namely those in which (Eα , Bα ) and (Pα /ρ0 , Mα /ρ0 ), respectively, were the independent constitutive variables. Of course, similar calculations can also be performed, if other sets of independent variables are chosen. They also lead to results similar to (6.4.21) relating the phenomenological coeﬃcients of the respective constitutive formulations. Below we shall present the transformations of all possibilities of the statistical and the Lorentz models (IV,V). On the other hand, the Chu-formulations (models I,II) and the Maxwell-Minkowski model (III) have been shown to be equivalent if in each of them the same functional dependencies for the free energy are taken. Hence, similar changes can also be performed for these formulations. In the remainder of this section we shall list the transformation rules for all these changes of dependent and independent constitutive variables; but we shall restrict ourselves to free energies of the complexity (6.4.18). Thus we write 1 1 (m) (m) Vα Vα + (Vα Vα )2 ψ= 2χ 4χ 2ρ0 4ρ0 1 1 (e) (m) + (Wα Wα )2 − 12 c(Θ − Θ0 )2 2 χ Wα Wα + 4χ 2ρ0 4ρ0 1 (m) 1 (e) + L Vα Vα (Θ − Θ0 ) + L Wα Wα (Θ − Θ0 ) 2ρ0 2ρ0 (6.4.22) 1 (m) (e) b1 Vα Vα + b1 Wα Wα Eββ −ν(Θ − Θ0 )Eαα + 2ρ0 1 (m) (e) b2 Vα Vβ + b2 Vα Vβ Eαβ + 2ρ0 1 λ(Eαα )2 + 2GEαβ Eαβ . + 2ρ0 Here, (Vα , Wα ) stands for the respective pairs of electromagnetic variables which will be chosen as independent variables. In particular, the following choices will be made: in the models I, II, III

in the models

IV, V

(Vα , Wα ) = (Eα , Hα ) :

ˇ· (ˇ)

(Vα , Wα ) = (Eα , Bα ) :

(ˇ·)

(Pα , MC α) :

ˆ· ) (ˆ

(Pα , ML α) :

(ˆ· )

(Eα , ML α) :

(·)

(Pα , Bα ) :

(·)

≈

(Eα , MC α) : (·) ++

(Pα , Hα ) : ( · )

∼

+

Correspondingly, ψ in (6.4.22) is the energy functional of the formulation at hand and must be characterized for each of these. The symbols used are

190

6 Linearization

also indicated in the above table; clearly, they must also be applied in all ˆˆ coeﬃcients on the right-hand side of (6.4.22). Recall, further, that ψˇ and ψ, for instance, are not the same energy functionals, but they are related to each other by Legendre transformations. We shall now list all the equivalence relations for the various formulations: (A): The transformations from the (ˇ·)- to the (ˆ· )-formulation have been listed in (6.4.21) and are labeled with the symbol (A). ∼

(B): The transformations between the constitutive theories (ˆ· ) and ( · ) are based on the energy functionals ML ψˆ = U − ηΘ + α Bα , ρ0 L M Pα ψ˜ = U − ηΘ + α Bα − Eα , ρ0 ρ0

(6.4.23)

and equivalence follows, if the following identities hold  µ 1  ˜(m) = 2 χ ˆ(m) = (m) , 2 χ ˜(e) = − (e) = −χ(e) ,  2χ   χ ˆ  2χ    (m) (m) (e) (e) 4 (e)  ˜ = 4χ ˆ , ˜ = (χ ) 4 χ ˆ , 4χ 4χ       ˜ (m) = L ˆ (e) , ˆ (m) , ˜ (e) = (χ(e) )2 L  L L ˜(m) = b ˆ(m) , b 1,2 1,2

˜(e) = (χ(e) )2 b ˆ(e) , b 1,2 1,2

cw , Θ0 ν˜ = νˆ = ν ,

˜=λ ˆ=λ, λ

˜c = ˆc =

˜=G ˆ=G, G ∼

               

(B) (6.4.24)

+

(C): To relate the ( · )- and ( · )-formulations one must start with the energy functionals Pα ML Eα , ψ˜ = U − ηΘ + α Bα − ρ0 ρ0

+

ψ= U − ηΘ ,

and then obtains the equivalence relations

(6.4.25)

6.4 Interchanging Dependent and Independent Constitutive Variables +(m)

+(e)

1 χ(m) , = − µ ˜(m) 2χ (m) 4 +(m) χ χ = ˜(m) , 4χ 4 µ (m) 2 + (m) χ ˜ (m) , L L = µ (m) 2 +(m) χ ˜(m) , = b b1,2 1,2 µ c + c = ˜c = w , Θ0

1 1 = (e) (e) ˜ χ 2χ 4 (e) + 1 ˜(e) 4χ 4 χ = χ(e) 2 + (e) 1 ˜ (e) , L L = χ(e) 2 +(e) 1 ˜(e) , = b b1,2 1,2 χ(e)

+

˜ =G, G= G

2

χ

=−

2

χ

=−

+

˜=λ, λ=λ +

ν = ν˜ = ν ,

191

  ,                                        

(C) (6.4.26)

+

(D): The energy functionals in the ( · )- and (ˇ·)-formulations have already been stated before, namely in (6.4.1) and (6.4.25)2 . Exploiting the identity + Pα ψˇ = ψ − Eα , ρ0 we obtain (m)

ˇ 2χ

(m)

ˇ 4χ

= =

+(m)

2

χ

+(m)

4

χ

+ (m)

ˇ (m) =L L

+(m) ˇ(m) =b b 1,2 1,2 +

ˇc = c= +

, ,

cw , Θ0

νˇ =ν = ν ,

=−

µ χ(m)

,

(e)

ˇ 2χ

1

=−

+(e)

2

,

(e)

ˇ 4χ

= (χ

= −χ

(e)

χ

(e) 4

)

+(e)

4

χ

+ (e)

ˇ (e) = (χ(e) )2 L L ˇ(e) = b 1,2

+(e) (χ(e) )2 b1,2

+

ˇ =λ = λ , λ +

ˇ =G = G , G

, ,

,

  ,                                        

(D) (6.4.27)

With these relations the constitutive theories of models IV and V are all compared. Performing the transformations (A), (B), (C) and (D) must lead to identical free energies. There remains to compare the constitutive theories for models I, II and III.

192

6 Linearization

ˇ ·- and ( ˆˆ· )(AA): We begin with the transformations for the (ˇ) formulations. In these (see (5.2.25) and (6.2.45)) we have Pα µ0 MC α ψˇ = U − ηΘ − Eα − Hα , ρ0 ρ0 ˆ ψˆ = U − ηΘ.

(6.4.28)

Note, however, that here U = II U , whereas in all the foregoing relations one must read V U for U . Comparing for both formulations the energy functionals of the complexity (6.4.22) we obtain the following relations: µ20 µ0 ˆ = (m) , ˆ(m) = − (m) 2χ ˇ χ χ 2 4 1 (m) ˆ ˇ(m) , = ˆ 4χ 4χ χ(m) 2 1 ˆ ˇ (m) , (m) ˆ L L = χ(m) 2 1 ˆ (m) ˇ(m) , ˆ b b1,2 = 1,2 χ(m) c w ˆ ˆc = ˇc = , Θ0 νˆ ˆ = νˇ = ν ,

 1 1  (e) ˆ ˆ = − (e) = (e) ,  2χ   ˇ χ  2χ   4   1 (e) (e)   ˆ ˇ  χ , ˆ = 4 4χ   χ(e)   2    1 ˆ  ˇ (e) (e) ˆ L L = , (e) χ (AA) (6.4.29)  2   1  ˆ ˇ(e) ,  ˆ(e) =  b b  1,2 1,2 (e)  χ     ˆ ˇ  ˆ  λ=λ=λ,       ˆ ˇ ˆ G=G=G,

(BB), (CC), (DD): It is now obvious how the transformations must look like for the remaining formulations. For the electrical coeﬃcients the transformations (AA), (BB), (CC), (DD) are identical to the corresponding ones indicated with a single letter, whereas for the magnetic coeﬃcients we must only replace the factor (µ/χ(m) ) in the one-letter transformations by (1/χ(m) ) to obtain the double-letter transformations. What remains is then the transformation of one formulation out of the group (IV, V) to one of (I, II, III). This problem has already been solved in ˇ ·-formulation is Sect. 5.7 in which the transformation from the (ˇ·)- to the (ˇ) ¯ For completeness we shall repeat the discussed. (There, ψˇ was denoted by ψ.) results here. We shall characterize this transformation, which makes the link between the two distinct groups (IV, V) and (I, II, III) and, therefore, completes the comparison of all possible constitutive theories, with the symbol A. It reads

6.5 Discussion

(A): ˇ(m) 2χ

= µ0 µ 2 χ ˇ(m) = −µ0 χ(m)

ˇ(m) 2χ

= µ4 4 χ ˇ(m) ,

ˇ (m) = µ2 L ˇ (m) , L ˇ(m) = µ2 b ˇ(m) b αβγδ αβγδ +µ0 χ(m) (2 + χ(m) )nαβγδ ˇc = ˇc = cW , Θ0 ˇ ν = νˇ = ν ,

 = 2χ ˇ(e) = −χ(e) ,       (e) (e) ˇ  = 4χ ˇ ,  4χ      ˇ (e) (e)  ˇ ,  L =L     (e) (e) ˇ ˇ bαβγδ = bαβγδ , (A)      ,        ˇ ˇ  λ=λ=λ,       ˇ ˇ G=G=G,

193

ˇ(e) 2χ

(6.4.30)

This completes the explicit comparison of all possible constitutive theories in the formulations (IV, V) and (I, II, III) as far as the free energy goes. A complete comparison must, however, also include the constitutive relations for the electric current and the heat ﬂux vector. They must in all formulations be the same functions. For relationships of the complexity of Sect. 5.2.3 this means that relations must hold such that (see (6.2.66)) ˇ ML γ ˆ γδ , − ∂ ψ , Θ) . (6.4.31) Λαβ = Λˇαβ (Eγδ , Bγ , Θ) = Λˆ Eγδ , , Θ = Λ(E ρ0 ∂Bγ These relations can be elaborated in an analogous way as above and thus lead (d) (d) to relations between coeﬃcients as, for instance, Λˇαβγδ and Λˆαβγδ etc. These are, however, at most second order eﬀects, and for most practical problems linear relationships of the form Qα (= QSα ) = −καβ Θ,β + βαβ Eβ , Jα =

1 βαβ Θ,β + σαβ Eβ , Θ

(6.4.32)

suﬃce. In (6.4.32), now καβ , βαβ and σαβ are constants which correspond to the tensor of heat conduction, the tensor of electrical conductivity and the tensor of the thermoelectric eﬀect. Since the temperature gradient and the Lagrangean electromagnetic ﬁeld are the same variables in all formulations, it suﬃces for a full equivalence of all theories to choose for καβ , βαβ , σαβ the same numerical values.

6.5 Discussion In this chapter, the governing equations of thermoelastic polarizable and magnetizable solids were subjected to a decomposition procedure, which allowed a separation of the general problem, described mathematically by a

194

6 Linearization

set of nonlinear partial diﬀerential equations, into two simpler problems. One of these involves as unknowns only small quantities, so that all products of these could justly be omitted. The emerging set of equations was linearized this way. We saw that linearization by itself turned out to be a fairly complex problem if not attacked with the proper formulation. The Lagrangean description treated in Chap. 5 provided the appropriate vehicle to avoid all complexities which arose in the Eulerian description. The decomposition of the governing ﬁeld equations into equations valid for the intermediate state and the perturbation state was performed for the Lorentz model only, because all theories were proved to be equivalent. Hence, and if one so desires, all problems can be solved just with this formulation. However, because all formulations are used in the current literature, the transformations of the phenomenological parameters from one theory to another one should be known, if the link between all these theories should explicitly be possible. With the results derived in Sects. 5.7 and 6.4 this can easily be achieved for the restricted constitutive class investigated there. All necessary informations to perform such transformations are contained in Fig. 6.1, in which the two circles stand for models (I, II, III) and (IV, V), respectively. At the periphery, we show those electromagnetic ﬁeld variables, which are considered as independent electromagnetic ﬁelds of the constitutive theory. The symbols used in these constitutive theories are also indicated in the ﬁgure. In Sect. 6.4 we derived the transformation rules necessary for the respective constitutive formulations to lead to equivalent electromagnetic theories. The transformations as listed in Sect. 6.4 are indicated in Fig. 6.1 by an arrow on the periphery connecting the two formulations to which they apply. The arrow is also characterized by a symbol (A), (B), . . . , (DD), indicating the transformation rules derived in Sect. 6.4 and valid for the transfer

Fig. 6.1. Scheme for the transformation of a constitutive formulation of any model into any other one

6.5 Discussion

195

of a formulation into the neighboring one on the periphery of the circle. It is evident that to every constitutive theory within a circle any other one corresponds. The missing piece to obtain full correspondence of all presented formulations is the double arrow connecting both circles and characterized by the symbol (A). The transformation from the (Eα , Hα )-formulation in models (I, II, III) to the (Eα , Bα )-formulation in models (IV, V) was derived in Sect. 5.7 and recapitulated in Sect. 6.4. It allows a connection between anyone of the constitutive theories in the models (I, II, III) with any other one in the models (IV, V). With this result the main objective is achieved. We must emphasize once more, however, that conditions of equivalence were derived above for free energy functionals, which were regarded to be truncated polynomial expansions and that equivalence was established to within terms omitted in the respective expansion procedures. Moreover, for reasons of transparency in the presentation and in order to avoid long manipulations, we restricted ourselves to isotropic solids for which equivalence conditions turned out to be simple formulas relating the phenomenological coeﬃcients. Of course, the calculations can be performed also for an anisotropic solid, but they are very long and, furthermore, equivalence formulas are complex. Apart from its practical signiﬁcance the study of the equivalence conditions has led us to a deeper theoretical understanding at least in the following regard: It is known that the phenomenological constants of the free energy as given in (6.4.15) and (6.4.16) bear standard names. These are as follows: (m) 2 χαβ

magnetic constants

(e) 2 χαβ

electric constants

(em) 2 χαβ

electromagnetic coupling constants electric, magnetic anisotropy constants

(e) (m) 4 χαβγδ , 4 χαβγδ (e)

(m)

(e)

(m)

pyroelectric, pyromagnetic constants

λα , λα

thermoelectric, thermomagnetic constants

Lαβ , Lαβ

piezoelectric, piezomagnetic constants

εαβγ , εαβγ

thermoelastic constants

(e)

(m)

ναβ

thermal constant

c

elastic constants

cαβγδ

The association of names with a certain phenomenological constant is not unique, however. In particular, we know that dependent on the constitutive approach the magnetic constants are called magnetic permeability or magnetic susceptibility. Moreover, we saw that for an isotropic body, in which there are no electromagnetic coupling terms and no pyro- and piezoelectric

196

6 Linearization

and -magnetic eﬀects, only the last three constants in the above list turned out to remain the same in all formulations. The transformation rules of all other constants involved the magnetic or the electric constants so that the deﬁnition of a particular eﬀect depended on the choice of the constitutive variables and on the formulation. For anisotropic bodies the situation becomes even worse, and indeed, one can show that in a body, in which the third-order constants eαβγ do not all vanish, not even the elastic coeﬃcients are unique. Similarly, in a material with nonvanishing λα ’s thermoelastic constants and the speciﬁc heats are not uniquely determinable either. Theoretically, these results are very important ones, because they say, for instance, that in an anisotropic piezomagnetic body the elasticity coeﬃcients depend on the formulation used. Strictly this means that in such a material the elasticity coeﬃcients cannot be determined from an experiment in the absence of electromagnetic ﬁelds. Practically, on the other hand, the corrections of the ﬁeld free elasticity coeﬃcients are so small that the eﬀect of the piezomagnetic constants on them can always be neglected. The constitutive theories developed in this monograph apply to all bodies which are called magnetizable and polarizable thermoelastic solids. By setting the appropriate phenomenological constants to zero, all special cases of it can also be derived. If, for instance, all coeﬃcients bearing the superscript (m) vanish, the body is called polarizable-only or dielectric. If on the other hand, all coeﬃcients with a superscript (e) vanish, the body is magnetizable-only. This deﬁnition is independent of the fact whether electrical conduction is present or not. Often, dielectric substances are electric insulators, however. Before we close, we would like to draw the readers attention on a few practical problems to which the theories presented in this book and in particular the decomposition procedure of this tractate may be applied. An extensive monograph on such problems is F.C. Moon’s “Problems in Magneto-Solid Mechanics”, [155], so that it suﬃces if we point out some of the more interesting problems. In principle the problems can be classiﬁed into two groups: ﬁrst, dynamical problems such as magnetoelastic wave propagation and vibration problems, and second, magnetically and electrically induced bifurcation problems. Both classes of problems have been attacked by several authors already, but so far this has only been done in the Eulerian description. Early applications to vibration and wave propagation are by Dunkin and Eringen [62], Kaliski and Petykiewicz [105], Paria [174], Alers and Fleary [11] and M.F. McCarthy [140]. The linearization procedure of these authors is, however, only sketchy and, as far as dynamical equations go the theories are quasi-static. A ﬁrst attempt to investigate the wave propagation problem with a linearized theory which proceeds along the lines of this article is due to Hutter, [93, 94]. But his equations are based on those of Hutter and Pao [91], which have been found not to be completely correct for the reasons explained in Sect. 6.1. Hence, the inﬂuence of (polarization and) magnetization on the propagation of waves in solids should still further be attacked,

6.5 Discussion

197

and experiments should be performed, which would corroborate or disprove the theoretical predictions. This has already be done to a certain extent by Moon and Chattopadhyay [154], but an extensive comparison with the results in [93, 94] was not attempted. Of similar nature are the vibration problems treated by Tiersten [234] and Van de Ven [248]. Both authors use a linearization procedure equivalent to that presented above, but using the Eulerian description and only for a quasi-static theory (in [249], however, it was shown that the results for a dynamic, non-relativistic theory correspond to those of the quasi-static case). Tiersten determines the eigenmodes of magnetically saturated plates of cubic Yttrium-Iron-Garnet in a large static transverse magnetic ﬁeld, perpendicular to which a small time-periodic ﬁeld is superimposed. Van de Ven, on the other hand, discussed vibrations of circular cylinders. First applications in the bifurcation of electro- and magnetoelasticity trace back to 1967 when Moon and Pao [152] presented experimental and ﬁrst theoretical results on the buckling of a soft ferromagnetic plate in a homogeneous magnetic ﬁeld. The discrepancies of the theory and of the experimental evidence initiated further work, notably by Wallerstein and Peach [260], Popelar [184], Dalrymple, Peach and Vliegelahn [51], Pao and Yeh [170] and Alblas [10], but with the exception of [170, 10], these articles do not make use of a proper linearization scheme. Alblas [10], includes in his analysis the buckling of circular rods based on a Liapunov approach. Since in all of the aforementioned articles the plate boundary conditions were not derived consistently, the entire matter was reinvestigated by Van de Ven [250], who also compared the existing formulations of static magnetoelasticity, in particular the Maxwell–Minkowski and the `re-current model, which in the static theory agrees with the Lorentz Ampe model. It turns out that, ultimately all models lead to the same buckling values, as they must; however, the reasons for this buckling can in each case be interpreted diﬀerently: (i) in the Maxwell–Minkowski formulation buckling is due to distributed magnetic surface forces at the upper and lower surface of the plate and a shear force per unit length of magnetic origin at the boundary of the plate; `rean current model, on the other hand, buckling originates (ii) in the Ampe from distributed surface moments of magnetic origin at the upper and lower surface of the plate; the boundary is now free of forces. The diﬀerence in the loading could be traced back to diﬀerences in the constitutive equations for the shear forces and bending moments in the plate. Other interesting bifurcation problems are the buckling of superconducting rings and coils which carry a large electric current. A ﬁrst step towards a solution was done by Moon and Chattopadhyay [153], but further investigations are still needed to improve their solution. A contribution to this ﬁeld of research is due to Alblas. These problems are of extreme practical interest in future reactor technology.

Part II

Applications Magnetoelastic (In)stability and Vibrations Electrorheological Fluids

7 Magnetoelastic (In)stability and Vibrations

7.1 Introduction In modern technological equipments, such as fusion reactors, magnetic energy storage devices, MRI-scanners, superconducting generators, and magnetically levitated trains, huge magnetic ﬁelds occur. These ﬁelds are generated by high currents through superconducting coils. In these devices, the superconducting currents are so high that the coils are subjected to strong magnetic forces. These forces can cause unwanted vibrations (resulting in loud unpleasant noise as in MRI-scanners) or even collapse (buckling) of structures (coils) in e.g. fusion reactors. Therefore, in the design of these devices the analysis of the vibrations and the stability of magnetic and superconducting structural elements due to electromagnetic forces plays an important role. In this chapter, we consider the (in)stability of (systems of) ferromagnetic bodies placed in an external magnetic ﬁeld and of superconducting structures loaded by Lorentz forces due to the electric currents in these conductors. Since a stability problem is always an essentially nonlinear problem, the theory for it must be built upon a nonlinear set of equations for a magnetoelastic model. As seen in the ﬁrst part of this book, several such models exist and thus one speciﬁc model must be chosen ﬁrst. After that the general approach to the problem could run as follows: the ﬁnal deformed state is considered as a perturbation of an intermediate state, for which in general the rigid-body state may be taken, and the ﬁelds in the deformed state are linearized with respect to the intermediate state. When the resulting homogeneous linear system has a non-trivial solution, we say that the intermediate state is unstable. We start this chapter with a historical review of magnetoelastic buckling problems, in which we recapitulate some earlier results from the literature. Section 7.3 deals with ferromagnetic systems. We introduce both the socalled classical method and a variational method. Both methods are illustrated by examples dealing with a cantilevered beam of (narrow) rectangular or elliptic cross-section and a set of two parallel rods. In Sect. 7.4 the buckling of superconducting structures is treated. A variational method is introduced and illustrated by examples such as sets of two parallel rods or two concentric or parallel rings. The results are compared with the results of the so-called direct Biot–Savard method. It turns out that the latter method delivers a lower bound for the critical buckling current, which however can deviate K. Hutter et al.: Electromagnetic Field Matter Interaction in Thermoelastic Solids and Viscous Fluids, Lect. Notes Phys. 710, 201–278 (2006) c Springer-Verlag Berlin Heidelberg 2006 DOI 10.1007/3-540-37240-7 7

202

7 Magnetoelastic (In)stability and Vibrations

substantially from the value obtained by the more exact variational method, when the slender rod-like superconductors are too close to each other. More results for superconducting structures such as helical and spiral coils are presented in Sect. 7.5. In the latter section a somewhat modiﬁed method is proposed in which the law of Biot and Savard is used to construct an admissible magnetic ﬁeld for the variational method; we call this approach the combined method. Finally, Sect. 7.6 deals with magnetoelastic vibrations of magnetoelastic or superconducting systems. Eigenfrequencies are determined both by a direct method (comparable with the classical method for buckling) and by a variational method (a generalisation of Rayleigh’s principle to include magnetoelastic interactions).

7.2 Historical Review of Magnetoelastic Buckling Problems Magnetoelastic buckling is a phenomenon in which an elastic structure becomes unstable (buckles) under electromagnetic loading. Such a structure can be, for instance, ferromagnetic or (super)conducting. The ﬁrst investigations of technical relevance in this ﬁeld are those of Moon [156]. He considered both ferromagnetic and conducting systems, the latter in cooperation with Chattopadhyay; see [152, 40]. A more fundamental theory of magnetoelastic stability was presented by Alblas in [9]. In this respect, also the works of Eringen [69] and of Goudjo and Maugin [76], who investigated the instability of ferromagnetic plates, should be mentioned. This subject was also studied by Van de Ven in [250]. Of more recent date is the paper of Zhou, Zheng and Miya [282], who looked at magnetoelastic buckling of ferromagnetic plates with regard to the safety of the ﬁrst walls, or blankets, of a fusion reactor. These plates are of ferritic stainless steel, and thus ferromagnetic, and the interactions with the magnetic ﬁelds in the reactor are so strong that magnetoelastic buckling comes into sight. In 1997, Zhou and Zheng [283] examined the magnetoelastic instability and the increase of natural frequency of a ferromagnetic plate in a magnetic ﬁeld. They developed a variational formalism by use of which they derived one general expression for the magnetic force that covers both the stability and the frequency problem at one time. Yang [273] considered a special subject, namely the buckling of a piezoelectric plate; he found a buckling load that was greater than the purely mechanical buckling load, implying that neglect of piezoelectric coupling would yield a conservative estimate of the buckling load. Moon and Pao [156] studied the buckling problem of a cantilevered ferromagnetic beam of narrow rectangular cross-section placed in a transverse magnetic ﬁeld B 0 . They found the magnitude of the buckling ﬁeld B0cr to be proportional to the (3/2)-power of the thickness-to-length ratio. This result, however, was in disagreement with their own experimental results. This

7.2 Historical Review of Magnetoelastic Buckling Problems

203

discrepancy started a discussion in the literature between several authors notably among Wallerstein, O’Peach, Bast, Popelar, Dalrymple, and Miya et al.; for the latter, see [143]–[145]; for a further list of references, see [252]. Many diﬀerent approaches were tried, but they did not really result in a much better agreement. The main reason for this disagreement seems to lie in the assumption of an inﬁnite width of the beam-plate. Although the beam-plate was much wider than high, the inﬁnite assumption did yield unacceptable results. Ultimately, a good explanation was found by Van de Ven [252, 253], who developed an analysis accounting for the ﬁnite width of the rectangular cross-section. The main lines and results of this analysis are presented in the Example in Sect. 7.3.1. Yabuno [271] investigated the bifurcation in buckling of a beam subjected to an electromagnetic force and he proposed a control method to stabilize the magnetoelastic buckling. Tani and Otomo, at an IUTAM-conference in Paris 1983, presented a paper on the magnetoelastic buckling of two nearby ferromagnetic panels [231]. As a follow-up, Tani and Van de Ven et al. cooperated on the magnetoelastic buckling of two parallel rods. They found a good correspondence between their theoretical and experimental results (obtained in the laboratory of Tani in Sendai, Tohoku University); see [255] or Sect. 7.3.3. This set of two parallel rods will serve as a standard example throughout this chapter. Ambartsumian and his coworkers reported on the magnetoelastic interactions in magnetic and current-carrying plates and shells placed in external magnetic ﬁelds; for a review of their earlier work, see [12]. Especially the magnetoelastic stability of current-carrying plates and shells was intensively studied in the school of Ambartsumian,Amb2, [14]; in this respect, we also refer to the work of K.B. and R.A. Kazarian [109] and Ovakimian [169] and, of more recent date, Mol’chenko [148, 149]. Vibrations and stability of two-layered magnetostrictive and of superconducting plates were studied by Bagdasarian et al. [18, 19]. In 1975, Chattopadhyay and Moon [40] were the ﬁrst to give a closed form solution for the buckling problem of a current-carrying elastic rod in its own ﬁeld. The stability of conducting strings in a parallel magnetic ﬁeld were studied by Wolfe [266, 267], and by Nowacki [164]. Chattopadhyay [41] showed, by numerical means, that a superconducting circular coil in its own ﬁeld is always stable, a result conﬁrmed by Van de Ven and Couwenberg in [254] via an analytical solution. In Moon’s book [156, Ch.5 and 6], a number of other related problems is presented, e.g. circular coils in transverse or toroidal external ﬁelds. For more realistic problems, however, structural problems of greater complexity must be studied. The ﬁrst example is [84] in which Hara and Moon studied the internal buckling of superconducting solenoid magnets. A special nomination here deserves the work of the group around Kenzo Miya at the University of Tokyo, a.o. [143, 226, 162, 280, 279], who investigated, over a period of more than two decades, theoretically and numerically (ﬁnite

204

7 Magnetoelastic (In)stability and Vibrations

element analysis) as well as experimentally the stability of ferromagnetic plates and superconducting coils. Their research was especially directed to the design of fusion reactors, speciﬁcally with regard to the stability aspects; see [147] and [56] for two survey papers. Miya presented already in 1982 together with Takagi et al. [146] a ﬁnite element analysis for the buckling of an eight-coil superconducting full torus. Moreover, together with Zhou and Zheng, he analysed the stability of a superconducting three-coil torus in [280, 279] and of a superconducting helix [284]. Impulsive buckling of cylindrical shells was experimentally investigated in cooperation with Nemoto et al. in [162]. In [282], Zhou, Zheng and Miya investigated the stability of the so-called ﬁrst walls in a fusion reactor. By means of a variational principle the magnetoelastic buckling value for these ferromagnetic plates was found. In our view, several of the approaches to magnetoelastic buckling mentioned above are rather ad hoc. Moreover, often a number of more or less ad hoc assumptions are needed to keep the analysis in hand. Therefore, we felt a need for a uniﬁed approach. To this end, Van de Ven with his coworkers Lieshout, Rongen and Smits constructed a variational principle speciﬁcally suited for the solution of magnetoelastic buckling problems for ferromagnetic and superconducting systems [121, 122, 123]. They opted for a variational method in the hope that this would serve as 1. a uniﬁed method in a clear formulation; 2. a straight way to a direct solution for the exact buckling value, if possible, or otherwise; 3. a solid basis for numerical computations of the buckling value if an exact analytical solution is no longer possible. Thus, Van de Ven et al. designed a standard tool for this class of problems. In a series of papers they presented solutions for systems reaching from rather simple ones, such as sets of two or more parallel rods [122, 124], or rings [218] to complex ones such as helical or spiral coils [257]. Evident advantages of this method are 1. once a deﬁnitive form for the variational principle is chosen, the remaining analysis can be done in an exact way by mere analytical means; 2. whenever the principle is used in an approximate sense, the order of the approximations and the conditions for which they are allowed can be explicitly speciﬁed. In the following sections, we shall present the derivation of this variational principle and its application to a number of magnetoelastic buckling problems.

7.3 Ferromagnetic Systems

205

7.3 Ferromagnetic Systems 7.3.1 Classical Method In this section, we consider the (in)stability of (systems of) ferromagnetic bodies placed in an external magnetic ﬁeld. Since a stability problem is always an essentially nonlinear problem, the theory for it must be built upon a nonlinear set of equations for a magnetoelastic model. As seen in the ﬁrst part of this book, several such models exist and thus one speciﬁc model must be chosen ﬁrst. Once this is done, the general approach to the problem could run as follows: Consider a ferromagnetic body B placed in an external magnetic ﬁeld B 0 . Due to the action of magnetic forces, B will deform to a slightly deformed intermediate state GI . In general, the precise deformation in this state is of minor relevance for the stability considerations. It is the stability of this intermediate state which we want to investigate: we say that B buckles when GI becomes unstable. To investigate the stability of GI we superpose a perturbation (in both the deformations and the magnetic ﬁelds) on GI leading to the ﬁnal state G for B. By assuming the perturbations to be small, we can linearize the nonlinear set of equations and boundary conditions, referring to state G, with respect to the perturbations. This results in a homogeneous linear set of equations, now referring to GI . Since the deformations in GI are always very small (and as said, not essential for the stability problem) we may replace in the linearized perturbed system the intermediate state by the rigid-body state (or undeformed state). The rigid-body ﬁelds have to be calculated ﬁrst, and once this is done, the perturbed system has to be solved. However, as this is a homogeneous system it will in general only have the trivial zero-solution. Only for a discrete set of real (eigen)values for B0 = |B 0 | the perturbed system will have a non-trivial solution (the eigenvalues are real because the underlying problem is conservative). The lowest of these eigenvalues is the buckling ﬁeld B0cr . We will refer to this approach as the classical method. To explain this approach in a more formal, mathematical, way, we use the following scheme: Every equilibrium state of a body B, inﬂuenced by an external magnetic ﬁeld B 0 , is governed by a set of equations and boundary conditions. Let us denote this set schematically by S [B(x), M (x), T (x), x; B 0 ] = 0 .

(7.3.1)

The symbol S encloses various diﬀerential operators; some of them act on the boundary. In the theory of stability two conﬁgurations are distinguished, namely the intermediate state GI , satisfying S 0 [B 0 (ξ), M 0 (ξ), T 0 (ξ), ξ; B 0 ] = 0 ,

(7.3.2)

206

7 Magnetoelastic (In)stability and Vibrations

where ξ ∈ GI is the position vector in the intermediate state, and the ﬁnal (perturbed) state G, which diﬀers only slightly from the intermediate state, is characterized by S [B 0 (ξ) + b(ξ), M 0 (ξ) + m(ξ), T 0 (ξ) + T 1 (ξ), ξ + u(ξ); B 0 ] = 0 , (7.3.3) where ξ + u(ξ) = x, and T 1 is the perturbed stress tensor. Here, the perturbations {b, m, T 1 , u} are supposed to be small. Subtracting (7.3.2) from (7.3.3) and neglecting terms that are of second order in the perturbations, we arrive at a system that is linear and homogeneous with respect to the perturbations. The derivation of this system however is rather complicated because 1. the original set (7.3.1) is nonlinear; 2. the boundary conditions enclosed in S refer to the deformed boundary ∂G of B; 3. the constitutive equations for the stresses and the magnetization inside the body are diﬀerent from those outside the body. In the sequel, the linearized homogeneous problem is denoted by S 1 [b(ξ), m(ξ), T 1 (ξ), u(ξ); B 0 ] = 0 ,

(7.3.4)

where all operators enclosed in S 1 act on the boundary of the intermediate state. The set (7.3.4) always has the trivial zero-solution as a solution, but we are only interested in those values of B0 = |B 0 | for which [b, m, T 1 , u] = 0, is a solution of (7.3.4). The thus-posed problem is an eigenvalue problem; the perturbations [b, m, T 1 , u] and the magnitude B0 of the magnetic ﬁeld play the role of the eigenvector and the eigenvalue, respectively. In the theory of magnetoelastic stability these eigenvalues are called buckling values. Of course, we are especially interested in the lowest buckling value. The eigenvalue problem is linear with respect to the perturbations, but the buckling appears in a nonlinear way, due to the nonlinear dependence of the magnetic forces and stresses on B0 . Although in many cases a simpliﬁcation in which the intermediate state is replaced by the rigid-body state is allowed, this simpliﬁcation does not make the nonlinear dependence on B0 of the eigenvalue problem less complicated. Generally it is impossible to solve the buckling problem directly from (7.3.4). Therefore, it is important to have at our disposal a procedure, intended to obtain a reliable approximation for the buckling value. Practical experience shows that in general variational principles guarantee reliable approximations. We will come back to this subject further on in this chapter. Moon and Pao [156] applied the classical method to the buckling problem of a cantilevered ferromagnetic beam of narrow rectangular cross-section placed in a transverse magnetic ﬁeld and they found a buckling ﬁeld proportional to the (3/2)-power of the thickness-to-length ratio. This result was

7.3 Ferromagnetic Systems

207

in disagreement with their own experimental results. An explanation for this discrepancy was found by Van de Ven [252, 253], who developed an analysis accounting for the ﬁnite width of the rectangular cross-section. The main lines and results of this analysis are presented in the following example. Example: Magnetoelastic buckling of a soft ferromagnetic beam of elliptic cross-section (classical method) This example is based on [252] and [253]. Consider a slender cantilever beam of length l having an elliptic cross-section with semi-major axis a and semi-minor axis b (a > b). This beam consists of soft ferromagnetic elastic material (with magnetic permeability µ 1) and is placed in a uniform external magnetic ﬁeld B 0 = B0 ey , directed along the minor axis of the ellipse (the y-direction). Our analysis will be based on the general formulation for a magnetoelastic body in interaction with an external magnetic ﬁeld. When applied to a slightly deﬂected beam, this general formulation is linearized with respect to the pre-buckled state, which here is identiﬁed with the rigid-body state (classical method). In this way, two problems are obtained: one for the rigid-body magnetic potentials (for an inﬁnitely long straight beam), and a linear one for the perturbed magnetic potentials. For the determination of the buckling value of B0 , we need the potentials inside the beam only, which are here denoted by Φ(0) for the rigid-body state, and ϕ, the perturbed potential. For the description of the magnetoelastic interactions the Maxwell–Minkowski model (Sect. 2.3) is used. For a beam of a narrow rectangular cross-section, {−a < x < a, − b < y < b, b/a 1}, a so-called beam-plate, this would yield the following expression for the normal stress on the wider upper plane y = b of the beam-plate (see e.g. [250]; this expression also follows straightforwardly from (compare with eq. (3.4.25)1 in Sect. 3.4 of this book) & µ2 ∂Φ(0) ∂ϕ && . (7.3.5) tyy = µ0 ∂y ∂y & y=b

Let the deﬂection of the slender beam in the y-direction be given by V (z), z being the axial coordinate of the beam; then the bending of the beam is governed by the one-dimensional beam equation EIy V iv (z) = q(z) ,

(7.3.6)

where EIy is the bending stiﬀness and q(z) the normal load per unit of length, Ty ds , (7.3.7) q(z) = Γ

with Γ the boundary of the elliptic cross-section, and Ty the traction on Γ in the y-direction given by (compare with (7.3.5))

208

7 Magnetoelastic (In)stability and Vibrations

Ty =

µ2 ∂Φ(0) ∂ϕ ny , µ0 ∂n ∂n

(7.3.8)

with n the unit normal vector on Γ . As demonstrated in [252], the rigid-body magnetic potential can be calculated by use of elliptic coordinates. However, the advantage of an elliptic cross-section is that the magnetic ﬁeld inside the beam is uniform; calculations in [252] reveal that this internal potential is given by Φ(0) = −

(1 + β) B0 y , µ

(7.3.9)

where β = b/a is the thickness-to-width ratio. In [252], the following boundary value problem for the internal, ϕ(x, y, z), and external, ψ(x, y, z), perturbed potential is derived: • in the vacuum outside the beam ∆ψ(x, y, z) = 0 ,

ψ→0,

as

|x| → ∞ ;

(7.3.10)

• inside the beam ∆ϕ(x, y, z) = 0 ; • at the boundary Γ ψ − ϕ = −µ

(7.3.11)

∂Φ(0) ny V (z) ; ∂n

and ∂ϕ ∂ ∂ψ −µ = ∂n ∂n ∂n

∂Ψ (0) ∂y

(7.3.12)

V (z) .

(7.3.13)

For the solution of this system, we introduce a separation of variables ansatz according to ψ = Ψ (u, v)V (z) , ϕ = Φ(u, v)V (z) , (7.3.14) where u, v are elliptical coordinates. This separation of variables is, however, only consistent with ∆ϕ = 0 provided that V (z) + λ2 V (z) = 0 ,

(7.3.15)

subject to V (0) = V (l) = 0 for the cantilever, yielding λ=

π , 2l

(7.3.16)

for the eigenvalue λ. With this separation of variables, the Laplace equation for ϕ(x, y, z), (7.3.11), transforms into the Helmholtz equation for Φ(u, v), ∆Φ(u, v) − λ2 Φ(u, v) = 0 .

(7.3.17)

7.3 Ferromagnetic Systems

209

The solution of the resulting system is quite complicated and needs the use of Mathieu functions. Therefore, we do not give the details here, but only refer to [252] for those readers who are interested in the mathematics of the solution procedure. Using the results of these calculations, we obtain the following expression for the load q(z): 2πB02 (1 + O(ε)) , (7.3.18) q(z) = µ0 κΛ where ε = πa/4l 1 for slender beams, κ = −γ − ln((1 + β)/2) − ln ε, γ = 0.5772, and Λ = 1 + (2βµε2 κ)−1 . Note that although ε 1, µ 1 so that µε2 can be strictly of order unity. Substitution of (7.3.15) into the beam equation (7.3.6) yields (with Iy = πab3 /4, for the elliptic cross-section) π 2πB02 Eab3 V iv (z) − V (z) = 0 . 4 µ0 κΛ

(7.3.19)

Taking into account (7.3.15) and (7.3.16), we infer that (7.3.19) has only then a non-trivial (non-zero) solution if B0 is equal to its critical or buckling value

B √ 0 µ0 E

= cr

1 √ εκΛ 2

πb 2l

3/2 .

(7.3.20)

Let us now consider two limiting cases for the elliptic cross-section, namely 1. the circle: a = b = R, (⇒ β = 1); 2. the very wide narrow ellipse: b a, (⇒ β 1). For both cases it is assumed that ε = πa/4l 1. For the circular rod this leads to * 2 2 B0 πR κ 1 2l 1 √ + = , 2 2 µ πR 2l µ0 E cr

(7.3.21)

where now κ = −γ−ln((1+β)/2)−ln(πR/4l). This expression is in agreement with the results of Moon [156, Sect. III.3], Alblas [9], or Van de Ven [251]. For the more interesting case of the very wide ellipse, we note that in the preceding derivations three dimension ratios are used, viz. b/a, a/l, and b/l, though only two are independent. Since we consider here b/l as the basic variable, there remain two possible choices for the second variable. We choose b/a or β as the second independent variable, to obtain * 2 2 πb κ 1 2l 1 B0 √ + = , (7.3.22) 2 2β µ πb 2l µ0 E cr where κ = −γ − ln((1 + β)/4β) − ln(πb/2l).

210

7 Magnetoelastic (In)stability and Vibrations

Assuming that µ is so large that the second term in the square root in (7.3.22) may be neglected, we see that this relation reduces to

B √ 0 µ0 E

= cr

1 2

+

κ 2β

πb 2l

2 .

(7.3.23)

Ignoring the b-inﬂuence in κ, we conclude that B0cr at ﬁxed β is proportional to the second power of b/l. This result is in conﬂict with the result of Moon and Pao in [152], and many others who followed the same approach and found a (3/2)-dependence of b/l. The basic reason for this diﬀerence is that Moon et al. assumed the narrow rectangular cross-section to be inﬁnitely wide, whereas we considered here an ellipse of ﬁnite extent. The crucial point is that the solution of the Helmholtz equation (7.3.17) for the perturbed potential is fundamentally diﬀerent for a cross-section of ﬁnite width from that of an inﬁnite width, no matter how small the ratio b/a is. As indicated in [253], the power 2 is also much better in agreement with the experimental results reported in [152] than the power 2/3 (see also Table 7.1 at the end of the next example). Our purely analytical method presented in [252] for an elliptic crosssection is not possible for a rectangular cross-section; however, in [253], we presented an approach based on a very reasonable approximation holding for a narrow rectangular cross-section of ﬁnite width yielding an analogous result as found in (7.3.23). In a second example below, we shall show how we can use our variational approach to ﬁnd the buckling value for a rectangular cross-section. Again, this result conﬁrms our result for the elliptical cross-section. This concludes our ﬁrst example. 7.3.2 Variational Method for Ferromagnetic Systems In the magnetoelastic stability theory, as presented in the preceding example, it has been the usual procedure to start from a linear set of equations for the perturbations and to look for a value of the basic ﬁeld parameter B0 for which this set has a non-trivial solution. Since an exact 3-dimensional solution is often very diﬃcult, and magnetoelastic buckling problems almost exclusively occur for slender bodies or structures, one starts looking for adequate approximate solutions. This is usually done in the following way: For a slender body the 3-dimensional displacement u is approximated by a 1- or 2-dimensional characteristic displacement w; this can be for instance the deﬂection of the central line of a beam or that of the central plane of a plate. This chosen w has to satisfy the global equilibrium equation, inclusive forces of magnetic origin, depending on B0 . The buckling value is then found as the ﬁrst eigenvalue for B0 for which this equation has a non-trivial (non-zero) solution.

7.3 Ferromagnetic Systems

211

However, as the solution obtained by the procedure described above is not an exact solution, but merely a reasonable approximation, the obtained value for B0 is also an approximation. Therefore, let us introduce the small parameter (0 < 1) as a measure for the approximation error in the perturbed displacement ﬁeld. Then it is evident that, due to the linear character of the perturbed equations, the error in the found eigenvalue for B0 is also of the ﬁrst order in . In this respect, use of a variational principle clearly has the advantage that in such a procedure the error in B0 is of second order in . This can be explained best by describing globally the main lines of a variational method. These lines are successively as follows: 1. Starting from a given functional L, which attains its minimum in the ﬁnal (deformed) state (and which is quadratic in the perturbations) one has 2. to choose a class of trial functions (for the perturbed magnetic and displacement ﬁelds) satisfying the constraints of the variational principle; 3. to determine the best members out of this class by equating the ﬁrst variations of L equal to zero; 4. to calculate the buckling value for B0 from the equation J = 0, where J is the second variation of L; see (7.3.32), further on. Due to the stationary behaviour of the quadratic functional J the deviation between the exact buckling value and the approximated one calculated as in point 4. above is of the order of the square of the deviation between the exact and the approximated perturbations. Hence, the accuracy of the variational method is one order higher than that of the classical method. The choice of a class of trial functions (point 2.) is usually based on the choice of a displacement ﬁeld. In practice, buckling theory always applies to slender bodies or structures, such as beams, rods, plates, rings, or more complex structural elements. For slender bodies, the displacement in buckling can be characterised by one or two global displacement parameters, such as the deﬂection of the central line of a beam, or the normal displacement of the central plane of a thin plate. Clearly, it is assumed here that the intermediate ﬁelds are known. However, in many practical problems the deformations in the intermediate state are small, and have a negligible eﬀect on the buckling value. In these cases, the intermediate state may be replaced by the so-called rigid-body state, which in general is more simple to determine. Let us proceed now with a more advanced explanation of the basic idea behind a variational principle for magnetoelastic buckling. Assume that the set S in (7.3.1) encloses equations and boundary conditions according to S = {S1 , S2 , ....Sk , ....}. Take now some of the equations and boundary conditions of (7.3.1) for granted, say Si [B(x), M (x), T (x), x; B 0 ] = 0, 1 ≤ i ≤ k , and analogously for (7.3.2)

(7.3.24)

212

7 Magnetoelastic (In)stability and Vibrations

Si0 [B 0 (ξ), M 0 (ξ), T 0 (ξ), ξ; B 0 ] = 0, 1 ≤ i ≤ k ,

(7.3.25)

and consider these equations as constraints for the variations of the functionals L [B(x), M (x), T (x), x; B0 ]dV , (7.3.26) L [B, M , T ; B0 ] = R3

L0 [B 0 , M 0 , T 0 ; B0 ] =

R3

L 0 [B 0 (ξ), M 0 (ξ), T 0 (ξ), ξ; B0 ]dV ,

(7.3.27)

respectively. The integrands, the so-called Lagrangean densities, are connected with the sets of equations and boundary conditions (7.3.1) and (7.3.2) and need to be speciﬁed later on. Evaluation of Si −Si0 = si and L−L0 in terms of the perturbations results in (7.3.28) si [b(ξ), m(ξ), T 1 (ξ), u(ξ); B0 ] = 0, 1 ≤ i ≤ k , and L − L0 = δL + J + O( 3 ) ,

(7.3.29)

where denotes the order of magnitude of the perturbations, δL is the ﬁrst variation of L with respect to the intermediate state, which contains only terms of order , while J is the second variation of L, containing terms of order ε2 only. If the Lagrangean L is chosen in such a way that δL = 0 ∧ Si0 [B 0 (ξ), M 0 (ξ), T 0 (ξ), ξ; B 0 ] = 0, 1 ≤ i ≤ k ,

(7.3.30)

is equivalent to (7.3.2), then it can be proved that δJ = 0 ∧ si [b(ξ), m(ξ), T 1 (ξ), uξ; B0 ] = 0, 1 ≤ i ≤ k ,

(7.3.31)

is equivalent to (7.3.4). For the proof, we refer to [121]. Hence, the eigenvalue problem for B0 is equivalent to ﬁnding a non-zero solution of (7.3.31). Using that J is a homogeneous quadratic functional in the perturbations, we infer the important property δJ = 0

=⇒

J =0.

(7.3.32)

The latter two results constitute a variational principle that should be used in the following way: First, a class of trial functions {b(ξ), m(ξ), T 1 (ξ), u(ξ)} is constructed, satisfying the constraints (7.3.28). Those trial functions, which approximate the exact perturbations best, are determined from δJ = 0. Subsequently, an approximation of the buckling value B0 is calculated from the equation J = 0; see (7.3.32). Let be the order of magnitude of the deviations between the exact and the approximate perturbations; then, due to the stationary behaviour of J, the deviation between the exact and approximate buckling value is of order 2 .

7.3 Ferromagnetic Systems

213

In the above analysis, the determination of the Lagrangean density L is rather diﬃcult. Therefore, we follow here the reverse way: a Lagrangean density accompanied by some constraints is postulated, and by variation of L a set of equations like (7.3.24) is derived. If this set turns out to coincide with one of the generally accepted models presented in Chap. 3 of this book, we conclude that the right choice for L has been made. For ferromagnetic elastic bodies the variational principle is based on the following expression for the Lagrangean density L : 1 1 (B 0 , B 0 ) , L = − µ0 (H, H) − ρU + 2 2µ0

(7.3.33)

where U = U (F, M ) is the internal (magnetoelastic) energy density, F (with components Fiα ) is the deformation gradient, M the magnetization, and B 0 the uniform external magnetic ﬁeld at inﬁnity. The pertinent constraints are (here G− = G, G+ and ∂G are the conﬁguration of the body B, that of the surrounding vacuum, and the boundary of G, respectively, while an upper index − or + stands for a value inside or outside the body, respectively) Bi+ = eijk A+ k,j ,

Mi+ = 0 ,

Bi− = eijk A− k,j ,

Tij = ρ

∂U , ∂Fiα

x ∈ G+ , ρ=

ρ0 , JF

x ∈ G− ,

− A+ i = Ai ,

x ∈ ∂G ,

Bi+ → B0i ,

|x| → ∞ ,

(7.3.34)

where Hi+ =

1 + B , µ0 i

Hi− =

1 − B − ρMi− , µ0 i

JF = det F .

(7.3.35)

To obtain the total Lagrangean L the density L must be integrated over both the (ferromagnetic) body (G− ) and the surrounding vacuum (G+ ). The requirement δJ = 0, taking into account the constraints given above, yields the well-known Maxwell–Minkowski model for magnetoelastic interactions (Model III in Chap. 3). What we need next is the second variation J of L (J = δ 2 L); since this is a cumbersome derivation, we refer for the resulting expression of J and the underlying derivation to [121]. Since in the equilibrium state, of which the stability we are investigating, the ﬁrst variation of L is zero, J must be of second order in terms of the perturbations on the equilibrium state. If we neglect higher-order terms, J becomes a homogeneous quadratic function in the perturbations. However, as the ﬁnal state is again an equilibrium state, also the ﬁrst variation of J must be zero, but for a homogeneous quadratic function J this implies that then J itself must be zero. Hence, δJ = 0 =⇒ J = 0, and this relation delivers us directly an explicit expression for the buckling ﬁeld B0cr . This can be seen as follows: by splitting up J into a magnetic part K,

214

7 Magnetoelastic (In)stability and Vibrations

which is proportional to B02 (K = B02 K) and an elastic part W , according to J = K − W = B02 K − W , we see that J = 0 yields (K and W are independent of B0 ) (7.3.36) B0cr = W/K . Hence, if we can calculate W or K, either exactly or in a suﬃciently accurate numerical approximation, the relation above immediately delivers the lookedfor buckling ﬁeld. Lieshout and Van de Ven et al. used this method to calculate the magnetic buckling ﬁelds for a cantilevered beam of rectangular cross-section (an exact solution was found) and a set of two parallel rods, both systems being soft ferromagnetic, placed in a transverse magnetic ﬁeld, [122, 123]; the ﬁrst problem is discussed in the following example, and the second one in Sect. 7.3.1. Example: Magnetoelastic buckling of a slender soft ferromagnetic beam (variational method) This example is based on [122] or [123]. Consider again a slender cantilever beam of length l and of arbitrary simply-connected cross-section. This beam is of soft ferromagnetic elastic material (with magnetic permeability µ 1) and is placed in a uniform external magnetic ﬁeld B 0 = B0 ey , directed along the y-direction. This is the same problem, at least for an elliptic cross-section, as in the preceding example, but now we solve the problem by means of our variational principle. For this variational principle we need an expression for J = B02 K − W in terms of the perturbed magnetic potentials ψ and ϕ and the displacement u. Here, W is the elastic energy due to u, and K is the scaled magnetic energy (in the perturbed state). To start with the elastic energy, we choose for u a displacement ﬁeld owing to the bending of a slender beam, according to the Bernoulli–Navier theory, u(x) = V (z)e2 − yV (z)e3 ,

(7.3.37)

where V (z) represents the deﬂection of the central line of the beam. This choice yields for W the well-known expression for the bending energy of a slender beam l 1 W = EIy (V (z))2 dz , (7.3.38) 2 0 where EIy is the bending stiﬀness of the beam. The derivation of an expression for K is not so straightforward. Therefore, we omit here all details and immediately give the result, for the derivation of which we refer to [122] or [123]1 (in the next section we will present a general derivation for K, but then for the case of a superconducting structure): 1

In [122] and [123], the potential ψ is deﬁned by b+ i = ψ,i , whereas we use here the commonly accepted deﬁnition h+ i = −ψ,i ; this means that the ψ in [122] and [123] must be replaced by −µ0 ψ in our notation

7.3 Ferromagnetic Systems

K=−

Bi ui ∂G

∂ 1 Bj uj dS . ψ− ∂n µ0

215

(7.3.39)

This result holds for slender beams (quadratic terms in the slenderness parameter ε are neglected) and for soft ferromagnetic materials (terms of order µ−1 are also neglected). In (7.3.39), B is the rigid-body magnetic ﬁeld and ψ the perturbed magnetic potential both for the external region (the reason that only the external ﬁelds appear in this expression is that the internal ﬁelds are order µ−1 smaller than the external ones). The perturbed potential ψ = ψ(x, y, z) has to satisfy the following constraints (here, G+ is the vacuum region external to the beam) ∆ψ(x, y, z) = 0 , ψ + Bi ui = 0 , ψ→0,

for x ∈ G+ , for x ∈ ∂G , for |x| → ∞ .

(7.3.40)

The problem for the 3-dimensional potential ψ is reduced to a 2-dimensional problem by the separation of variables ψ(x, y, z) = Ψ (x, y)V (z) .

(7.3.41)

This separation of variables is only then consistent with ∆ψ(x, y, z) = 0, if V (z) satisﬁes the relation V (z) + λ2 V (z) = 0 ,

(7.3.42)

where the real parameter λ, the separation constant, is related to the length l through the support conditions; for a cantilevered beam λ = π/2l; see (7.3.15) and (7.3.16). With all this, (7.3.40) reduces to the 2-dimensional problem for Ψ (x, y) (here, D+ is the 2-dimensional region of the vacuum in the plane of the cross-section, and Γ is its boundary) ∆Ψ (x, y) − λ2 Ψ (x, y) = 0 , Ψ + By = 0 , Ψ →0,

for (x, y) ∈ D+ , for (x, y) ∈ Γ , for x2 + y 2 → ∞ .

(7.3.43)

Moreover, substitution of (7.3.41) changes the 2-dimensional integral in (7.3.39), with use of (7.3.37) for u, into a product of two 1-dimensional integrals according to l ∂ 1 K= − (Ψ − By By )ds V 2 (z)dz . (7.3.44) ∂n µ0 Γ 0 The problem (7.3.43) for Ψ can be solved by use of a conformal mapping: z = x + iy = h(ζ), which maps D in the xy-plane onto the unit circle in the ζ-plane. The transformed problem can then be solved by use of the theory of complex functions. However, we do not need a full solution for Ψ but rather

216

7 Magnetoelastic (In)stability and Vibrations

a calculation of the integral for K in (7.3.44). How this is done in detail can be found in [122]; we restrict ourselves here to giving only the result; it reads K=

1 π , µ0 κ(εc)

(7.3.45)

with κ = −γ − ln(εc), and ε is the slenderness parameter, as in the preceding example, while c is a constant that follows from the conformal mapping as c = limζ→∞ h (ζ). We note here that for the elliptic cross-section: ε = πa/4l and c = (1 + β)/2, in agreement with the preceding example. Substituting this into the equation for the buckling value (7.3.36), we arrive at 2 + κ πb B0 1 √ = , (7.3.46) 2 2β 2l µ0 E cr a result identical to (7.3.22) from the ﬁrst example. So our variational approach conﬁrms the result of the classical method. With the variational method, we could also solve analytically the buckling problem for a beam having a (narrow) rectangular cross-section. As can be found in [122] this leads to an expression for c in terms of elliptic integrals. We can calculate this value of c for all types of rectangular cross-sections, ranging from a square to very narrow ones. In [122], we compared the results with those for corresponding ellipses, and we found that the diﬀerence nowhere became more than 2%. With ‘corresponding’ we mean here that the elliptic and rectangular cross-sections do have identical thickness-to-width ratio’s and moments of inertia Iy . In (7.3.46), we have expressed B0cr in terms of the two dimension ratios β and b/l; instead of β we could also have used ε, which would yield (with β = b/a = 2ε(2l/πb))

B √ 0 µ0 E

= cr

1 √ κε 2

πb 2l

3/2 .

(7.3.47)

Hence, whether B0cr is proportional to b/l to the power 2 or 3/2 depends on the choice of β and b/l or ε and b/l as independent dimension ratios, respectively. However, as we shall show further on, experimental data of a.o. Moon and Pao [152] ﬁt more accurately to a second-power law than to a (3/2)-power law; see Table 7.1. Some further conclusions that can be drawn from the latter two results, (7.3.46) and (7.3.47), are: 1. At ﬁxed thickness-to-length ratio b/l the buckling value B0cr increases with increasing width-to-length ratio ε. This statement received experimental conﬁrmation from Dalrymple et al.; see [50, Fig. 4]. In this ﬁgure the variation of the buckling ﬁeld with the plate width a (at constant thickness b and length l) is shown. This curve corresponds very well with our result (7.3.47).

7.3 Ferromagnetic Systems

217

2. At ﬁxed thickness-to-length ratio b/l the buckling value B0cr decreases with increasing thickness-to-width ratio β. Again this statement is supported by experimental results from the literature; this time from Miya et al.; see [144, Fig. 3]. In addition, we note that the results of this paper also support statement 1. We conclude this example by giving a table in which for ﬁxed values of β and varying (b/l) our theoretical buckling values are compared with the theoretical and experimental results of Moon and Pao [152, Fig. 6]; see Table 7.1. This table shows good agreement between the theoretical values predicted by (7.3.46) and the experimental data from Moon and Pao. This agreement is so evident that we are convinced that this justiﬁes our conclusion that the discrepancy between the theoretical and experimental results of Moon and Pao is substantially due to the inadequacy of the inﬁnite-width assumption. The results of this example evidently reveal that the inﬂuence of the ﬁnite width of the beam is essential and remains so even for very small thicknessto-width ratios. We continue this section with one extra example. Table 7.1. Theoretical and experimental buckling values for a rod of rectangular cross-section

b l

× 103 3.0 2.5 2.0

√B0 µ0 E

β = 5.6 × 10−2 and µ = 6 × 104 cr (7.3.46)

0.96 0.69 0.46

√B0 µ0 E

cr [152],Th

1.87 1.42 1.02

√B0 µ0 E

cr [152],Exp

0.90-1.00 0.65-0.70 0.45-0.50

β = 2.4 × 10−2 and µ = 104 1.80 1.50 1.15 0.90

0.55 0.40 0.26 0.18

0.87 0.66 0.44 0.31

0.51 0.30-0.33 0.24 0.17-0.20

7.3.3 Magnetoelastic Buckling of a Set of Two Soft Ferromagnetic Parallel Rods Consider a system of two identical parallel slender soft ferromagnetic rods with radii R, length L, R/L 1, placed in an external magnetic ﬁeld B 0 acting in a plane normal to the axes of the rods, i.e. the Z-direction. A normal cross-section of the system in the XY -plane is given in Fig. 7.1. The distance between the centres of the rods is 2d. The uniform external ﬁeld B 0 lies in the

218

7 Magnetoelastic (In)stability and Vibrations Y

2d

X

O R

R

B0 α

Fig. 7.1. Cross-section of system of two parallel rods

XY -plane and makes an angle α with the positive X-axis. We will present the solution for the buckling problem for this system in two ways: in the classical way and by means of our variational principle. These presentations are based on [255] and [122], respectively. The classical procedure is almost analogous to that for one single beam. Here, we introduce the displacements of the central lines of the two cantilevered beams by the complex displacement w(m) (z), m = 1, 2, w(m) ∈ C of the m-th beam by w(m) (z) = u(m) (z) + iv (m) (z) ,

(7.3.48)

where u(m) and v (m) are the displacement in the x and y-directions, respectively, and z is the (real) axial coordinate. With this, the beam equation for the m-th beam becomes EI

d4 w(m) = q (m) (z) , dz 4

(7.3.49)

where I = πR4 /4 and q (m) is the magnetic load per unit of length on the beam. These loads are related to the rigid-body ﬁelds and the perturbed ﬁelds, and in order to ﬁnd the q (m) ’s one has to determine these ﬁelds. However, we refrain from doing this here, and refer for the details of the calculations to [255]. It turns out that the q (m) ’s are of the form

7.3 Ferromagnetic Systems

q (1) (z) = B02

219

Q1 w(1) (z) + Q2 w ¯ (1) (z) + Q3 w(2) (z) + Q4 w ¯ (2) (z) , (7.3.50)

while q (2) is obtained from q (1) by replacing the upper indices (1,2) by (2,1). In (7.3.50), w ¯ (m) is the complex conjugate of w(m) . However, Q1 to Q4 are complex numbers, which are explicitly determined by the solutions of the rigid-body and perturbed ﬁelds as can be found in [255]. The analysis presented there makes use of complex function theory and ultimately expresses the perturbed solution in terms of Bessel functions. We note here that these numbers depend on R, L, d and α, but are independent of B0 . The buckling equation (7.3.49) together with the separation condition (compare with (7.3.15) or (7.3.42))

w(m) (z) + λ2 w(m) (z) = 0 ,

λ=

π , 2L

m = 1, 2 ,

(7.3.51)

yields two homogeneous equations for w(1) and w(2) . For a cantilever support at z = 0, the boundary conditions are w(m) (0) = w(m) (L) = 0, from which the value for λ actually follows. It turns out that they can be decomposed into two independent equations for (w(1) + w(2) ) and (w(1) − w(2) ). Hence, two possible solutions exist: 1. w(1) = −w(2) : symmetrical buckling mode; 2. w(1) = +w(2) : anti-symmetrical buckling mode. The numerical results of [255] reveal that the lowest buckling value for B0 is associated with the symmetrical buckling mode. The corresponding buckling value reads + EI 2 , Ω = Re(Q1 − Q3 ) + |Q2 − Q4 |2 − (Im(Q1 − Q3 ))2 . B0cr = λ Ω (7.3.52) This buckling value depends on α and d/R; explicit results for B0cr , scaled with respect to the buckling value for one beam (d/R → ∞), as function of d/R are depicted in Fig. 7.2 for α = 0 and α = π/2. Discussion. In [255] also the pre-buckling deﬂections are calculated. Although it turns out that, as expected, these deﬂections are small, they nevertheless have an eﬀect that can explain the discrepancy between the theoretical and experimental results observed in the left graph of Fig. 7.2. For this, we consider the two cases α = 0 and α = π/2 one by one: 1. In case α = 0 both the pre-deﬂection and the buckling deﬂection are in the same plane. Hence, if the system is imperfection sensitive the pre-deﬂection will lower the buckling value. This explains why we experimentally found a somewhat lower buckling value than theoretically predicted.

220

7 Magnetoelastic (In)stability and Vibrations

Fig. 7.2. Buckling magnetic ﬁeld, scaled to the buckling ﬁeld for one rod, for symmetric buckling mode vs. distance between rods, scaled to radius; solid lines are theoretical values and and ◦ are experimental values. The left panel refers to a magnetic ﬁeld parallel to the plane of the rods (α = 0); the right to a transverse ﬁeld (α = π/2); from [255]

2. In case α = π/2 the plane in which the pre-deﬂection takes place is perpendicular to the plane of buckling. Hence, in this case there is no inﬂuence of the pre-deﬂection on the buckling value and, therefore, a much better correspondence between the theoretical and experimental results is observed in the right ﬁgure. We proceed by presenting a solution for the same problem, but now via the variational principle. For this, we start with the following approximate assumptions: 1. We neglect the inﬂuence of the intermediate deformations (see the Discussion above). 2. The magnetic susceptibility χ = 1 + µ is so high that all terms of order χ−1 are neglected. 3. All terms of order δ 2 are neglected, where δ = λR = πR/2L, is the slenderness parameter of the rods. 4. The rods deﬂect according to Bernoulli’s classical bending theory; the angle between the direction of bending and the X-axis is called θ1 , and the magnitude of the displacement of the center line is w(z) = w(1) (z) = −w(2) (z) (symmetrical buckling mode). 5. The rigid-body (intermediate) ﬁelds are independent of z (for long slender rods). 6. For the perturbed magnetic ﬁelds a separation of variables with respect to (x, y) and z, and proportional to w(z) can be applied (compare with the example of one rod). 7. Besides δ = λR, also λd = δd/R is small. The variational principle is now applied in the following way: ﬁrst, L − L0 is developed in the perturbations up to and including the second order, thus yielding J. This expression for J is simpliﬁed using the approximations listed

7.3 Ferromagnetic Systems

221

above. Next, a class of perturbations {b, m, T 1 , u} is chosen in such a way that the constraints (7.3.34) and the support conditions for the cantilevered rods are satisﬁed. We assume a symmetrical buckling mode, meaning that the displacement of the center line of the ﬁrst rod is given by u = V (z) cos θ1 ex + V (z) sin θ1 ey ,

(7.3.53)

where V (z) is the total deﬂection of the rod in the θ1 -direction. Then the elastic energy W is equal to that given in (7.3.38). For the magnetic functional K we obtain a formula analogous to (7.3.44), namely K = K1 (θ1 )K2 , 1 ∂ (Bx cos θ1 + By sin θ1 ) K1 (θ1 ) = Ψ + Bx cos θ1 + By sin θ1 ds , µ0 Γ1 ∂n L V 2 (z)dz , (7.3.54) K2 = 0

where Ψ = Ψ (x, y) is the perturbed external magnetic potential after a separation of variables analogous to (7.3.41). The calculation of the rigid-body ﬁelds Bx and By and the potential Ψ is done in [122] by using complex function theory together with a conformal mapping of the two circular cross-sections of the rods in the XY -plane onto an annular domain in the complex plane. Using Hilbert theory, we can then derive expressions for Bx , By and Ψ . However, it should be emphasized here that for our variational principle we do not need the explicit expressions for these variables, but we only need an explicit expression for the integral for K in (7.3.54). After several mathematical manipulations, for which we refer to [122], the following expression for J = W − KB02 , which depends on θ1 only, ,l is obtained (here, we have normalized the integral 0 V 2 (z)dz to unity): J(θ1 ) =

π 4 2π 2 Eδ − B Q(θ1 ) , 8 µ0 0

(7.3.55)

where Q(θ1 ) = c0 + c1 cos 2θ1 + c2 sin 2θ1 ,

(7.3.56)

and c0 , c1 and c2 are known functions (calculated in [122, eq. (5.13)]) of the parameters d/R and α. The optimal value θ1 = θˆ1 , corresponding to a minimum value of J, is obtained from dJ ˆ (θ1 ) = 0, dθ yielding

and

d2 J ˆ (θ1 ) > 0 , dθ2

c2 tan 2θˆ1 = . c1

(7.3.57)

(7.3.58)

The buckling value for B0 then follows from J(θˆ1 ) = 0, and this results in

222

7 Magnetoelastic (In)stability and Vibrations

θˆ1

B0cr

d/R

d/R

Fig. 7.3. Optimal angle for the deﬂection θˆ1 (left) and the buckling ﬁeld B0cr , scaled with respect to that for one rod (right) for diﬀerent values of α; from [122]

* B0cr =

µ0 E 16Q(θˆ1 )

πR 2L

2 .

(7.3.59)

Numerical evaluations show that the result (7.3.59) is equal to the previous result (7.3.52), at least for large values of the magnetic susceptibility (µ1 ≈ 0). Some of the numerical results are depicted in Fig. 7.3: the left panel shows θˆ1 as a function of R/d for a set of values for α; the right panel displays B0cr , scaled with respect to the buckling value for one rod, as a function of R/d for three values of α. From these graphs, we draw the following conclusions: 1. For ﬁxed values of R/d and α, the buckling value is proportional to ' (2 δ 2 = πR . 2L 2. An increase in R/d for ﬁxed values of α results in a decrease of the buckling value, whereas an increase in α for a ﬁxed value of R/d causes an increase of the buckling value. 3. For 0 < α < π/2, the angle of deﬂection θˆ1 is a decreasing function of R/d. 4. For 0 < α < π/2, the deﬂection is not in the direction of the ﬁeld (θˆ1 = α); only for α = 0 or α = π/2 one obtains θˆ1 = α. This concludes the section on ferromagnetic bodies; in the ensuing section we shall discuss instabilities in superconducting structures.

7.4 Superconducting Structures

223

7.4 Superconducting Structures Superconducting coils are extensively used in modern technology, as for instance in magnetic fusion reactors, NMR-scanners, magnetically levitated trains, and many other applications. The current in these devices is often so high that the structures carrying the currents are subjected to strong magnetic forces, which can become so large that the structure collapses (buckles). Hence, in the design of these devices investigation of the stability of the structure is of eminent importance, see e.g. [56, 147]. In reactors and scanners complex structures such as spirals, helical or toroidal coils are in use. The complexity of such structures makes the study of their stability very complicated, see e.g. [280, 284]. These methods often use approximations, of which the degree of accuracy could not always be indicated. To overcome this, we have adapted our variational method for ferromagnetic bodies as presented in the preceding section to one for superconducting structures [218]. In conducting systems, the electromagnetic loading is mainly due to the Lorentz forces, originating from an interaction of the electrical current with either its own magnetic ﬁeld or an external ﬁeld. Whenever these Lorentz forces become too strong, the structure buckles. The analysis of this (in)stability phenomenon can be performed analogously to the perturbation method presented schematically at the beginning of the preceding section. These methods often use approximations, of which the degree of accuracy could not always be indicated. To overcome this, we have adapted our variational method for ferromagnetic bodies as presented in the preceding section to one for superconducting structures [218]. To make the variational method of Sect. 7.3 suitable for systems of superconducting coils carrying a prescribed current I0 , the Lagrangean density must be changed into 1 L = (B, B) − ρU , (7.4.1) 2µ0 which is formally a Legendre transformation of (7.3.33) (for nonmagnetizable bodies) in which we pass from the variable H to B; see [218]. Since the total current I0 through the coil is prescribed, B must satisfy as a `re law constraint the Ampe (B, ds) = µ0 I0 , (7.4.2) C

where C is a contour encircling the conductor. The further evaluation is completely analogous to that in Sect. 7.3 and eventually amounts to (note that here the electromagnetic term is proportional to I02 ) (7.4.3) J = I02 K − W = 0 , =⇒ I0cr = W/K , yielding an explicit expression for the buckling current I0cr .

224

7 Magnetoelastic (In)stability and Vibrations

For not too complicated systems, the values of W and, above all, K can be calculated analytically or numerically, but both exactly, see [122] or [218]. When this is no longer possible, one can choose, on a variational basis, an appropriate (approximate) set of admissible ﬁelds to calculate W and K, and thus I0cr ; see [257]. A more detailed derivation of this variational principle is presented in the next section. 7.4.1 Formulation of Variational Principle for Superconducting Structures In this section, we shall start from expression (7.4.1) and derive a formulation for J = 0, where J is the second variation of L. We shall do this for a superconducting body (in practice, a slender structure) in vacuum. We denote by G the conﬁguration of the body, and by G+ that of the vacuum, while ∂G is the boundary of G. We model a superconducting body as a non-magnetizable body, for which the current density J (Amp`ere/m) is concentrated on the surface of the body, and for which the magnetic ﬁelds inside the body vanish (B − = H − = 0). The current density J is related to the boundary value of the external ﬁeld B + by µ0 J = n × B + ,

x ∈ ∂G .

(7.4.4)

We restrict ourselves here to a simply-connected superconducting body in a static situation. For our magnetoelastic model we will use the Maxwell– Minkowski model (Model III in Chap. 3). Moreover, since the internal ﬁelds are zero, we shall simply denote the external ﬁelds by B and H (without the superindex +). For the choice of L like in (7.4.1), the magnetoelastic equations for the ﬁnal state x in the Maxwell–Minkowski formulation are obtained by variation of L (i.e. δL = 0) under the following constraints: x∈G,

ρJF = ρ0 ,

x ∈ G+ ,

eijk Bk,j = 0 ,

∂U Fjα , ∂Fiα B → 0 , |x| → ∞ ,

Tij = ρ

(7.4.5)

plus an extra constraint prescribing the total current I0 by means of `re’s law (7.4.2). Ampe To ﬁnd the ﬁrst and second variation of L, we develop L in terms of the perturbations with respect to the intermediate state. In doing this, we take for the intermediate state the rigid-body state. The displacement u is then a perturbed ﬁeld, while we split the total magnetic ﬁeld as B = B 0 + b, where B 0 is the ﬁeld in the rigid-body state, and b is the perturbed ﬁeld. Developing L up to second-order terms in the perturbations, we obtain (J = δ 2 L)

7.4 Superconducting Structures

225

L − L0 = δL + J 1 1 = bi Bi dV − Bk Bk ui dS + Tij,j ui dV µ0 G+ 2µ0 ∂G G Tij Nj ui dS − ∂G 1 bk Bk ui + 12 Bk,j Bk ui uj + 14 Bk Bk (ui uj,j − uj ui,j ) Ni dS − µ0 ∂G 1 1 + bi bi dV − cijkl ui,k uj,l dV , (7.4.6) 2µ0 G+ 2 G where cijkl is the classical, Hookean, linear elasticity tensor. Since the constraints (7.4.5) and (7.4.2) have to be satisﬁed for both the intermediate and the present state, the constraints for the perturbations become X∈G, X ∈ G+ , and

tij = −Tij uk,k + Tik uj,k + cikjl uk,l ; eijk bk,j = 0 , b → 0 , |X| → ∞ ;

(7.4.7)

(b · ds) = 0 .

(7.4.8)

C

The constraints (7.4.7)2 and (7.4.8) for b guarantee the existence of a continuous magnetic potential ψ(X) for X ∈ G+ , such that bi = ψ,i ,

X ∈ G+ .

(7.4.9)

To dispose of irrelevant constants in ψ, we replace the last constraint of (7.4.7) by ψ → 0 , |X| → ∞ . (7.4.10) With this we can eliminate b from (7.4.6) in favour of ψ to obtain 1 1 δL = Tij,j ui dV + ψ,i Bi dV − Bk Bk Ni ui dS , Tij Nj ui + µ0 G+ 2µ0 G ∂G (7.4.11) and 1 1 J =− cijkl ui,k uj,l dV + ψ,i ψ,i dV 2 G 2µ0 G+ 1 ψ,k Bk ui + 12 Bk,j Bk ui uj + 14 Bk Bk (ui uj,j − uj ui,j ) Ni dS . − µ0 ∂G (7.4.12) By standard procedures, for which we refer to [122], we can prove that δL = 0 and δJ = 0 yield successively the equations for the intermediate

226

7 Magnetoelastic (In)stability and Vibrations

(rigid-body) and the perturbed (linearized) state for the magnetoelastic problem of a superconducting body in vacuum in the Maxwell–Minkowski formulation. In this respect, we note here that the rigid-body ﬁeld obtained from δL = 0 satisﬁes, besides the constraints (7.4.5)2 , also Bi,i = 0 , X ∈ G+ ,

Bi Ni = 0 , X ∈ ∂G .

For the further evaluation of J we ﬁrst conjecture that 1 1 − ψ,k Bk ui Ni dS = − (Bi,j uj − Bj ui,j )ψNi dS , 2µ0 ∂G 2µ0 ∂G

(7.4.13)

(7.4.14)

a relation which, with the aid of (7.4.13), can be proven as follows (−ψ,j Bj ui +ψBi,j uj −ψBj ui,j )Ni dS = eijk (eklm ψBl um ),j Ni dS = 0 , ∂G

∂G

(7.4.15) because the integrand in the last integral is a tangential derivative. Secondly, we note that for a pre-stressed linear elasticity problem and with the classical Hookean relation for cijkl , one has ν E ekk ell + ekl ekl , cijkl ui,k uj,l = Tjk ui,j ui,k + (7.4.16) 1 + ν 1 − 2ν where 2ekl = uk,l +ul,k . As the third step, we use Gauss’ divergence theorem to evaluate ψ,i ψ,i dV = ψψ,i Ni dS − ψψ,ii dV . (7.4.17) G+

∂G

G+

Finally, to dispose of the integral over the inﬁnite region G+ in the result above, we impose the extra constraint ψ,ii = ∆ψ = 0 ,

X ∈ G+ .

(7.4.18)

We note that, as the constraints do not prescribe ψ at ∂G, there is still freedom left for the variation of ψ. Using the four steps above, we obtain J in the form ν E 1 ekk ell + ekl ekl dV Tjk ui,jk ui dV − J= 2 G 2(1 + ν) G 1 − 2ν 1 − (ψ + Bk uk ),j Bj ui + 12 Bk Bk (ui uj,j − ui,j uj − uj,i uj ) 2µ0 ∂G − 12 Bj Bk (uj,k + uk,j )ui + (ψ,i + Bi,j uj − Bj ui,j )ψ Ni dS . (7.4.19) Here, the ﬁrst integral on the right-hand side is the contribution of the prestresses Tjk to the elastic energy, the second term is the classical (Hookean)

7.4 Superconducting Structures

227

linear elastic energy due to the perturbed deformations and the last integral is the perturbed magnetic interaction energy. The buckling current follows from J = 0; in order to obtain a direct expression for I0cr we scale the magnetic variables with respect to I0 . For this we introduce the following dimensionless ﬁeld variables: 2 1 ˆi = 2πR Bi , ψˆ = 2π ψ , Tˆij = (2πR) Tij , u B ui , (7.4.20) ˆi = 2 µ0 I0 µ0 I0 µ0 I0 R

where R is some length parameter that has to be chosen for a speciﬁc problem under consideration. With use of this in (7.4.19), the buckling equation J = 0 yields (we subsequently omit the hats) + + I0 µ0 W , (7.4.21) = E 2πR cr K where W is the scaled elastic energy 1 ν W = ekk ell + ekl ekl dV , 2(1 + ν) G 1 − 2ν

(7.4.22)

and K is

1 [(ψ + Bk uk ),j Bj ui + 12 Bk Bk (ui uj,j − ui,j uj − uj,i uj ) K=− 2 ∂G − 12 Bj Bk (uj,k + uk,j )ui + (ψ,i + Bi,j uj − Bj ui,j )ψ]ni dS (7.4.23) + 12 Tjk ui,jk ui dV . G

The ﬁnal result (7.4.21)–(7.4.23) immediately gives the buckling current I0 once we know ψ and u, either exactly or approximately. Assuming that the buckling deﬂection is in the x-direction, the external potential Ψ has to satisfy the constraints: ∆ψ(x, y, z) = 0 , ∂ (ψ + Bi ui ) = 0 , ∂n ψ→0,

for x ∈ G+ , for x ∈ ∂G ,

(7.4.24)

for |x| → ∞ .

We now illustrate the variational method on the example of a set of two parallel superconducting rods. Example: Buckling of two parallel superconducting rods (variational method) We consider a set of two identical parallel inﬁnitely long superconducting circular rods of radius R. The distance between the centres of the two rods

228

7 Magnetoelastic (In)stability and Vibrations

is 2d. The rods are periodically supported over a distance L. The currents through the rods have magnitude I0 and are in the same direction. Axes X, Y, Z are chosen as in Fig. 7.1. The dominant buckling mode turns out to be symmetric and in the X-direction; this means that the deﬂection of the second rod is equal but opposite to that of the ﬁrst rod. For the ﬁrst rod, we choose the displacement ﬁeld (appropriate for slender rods in bending) as u1 = V (z) ,

u3 = −xV (z) .

u2 = 0 ,

The elastic energy is then, as in (7.3.38), L π 4 W = ER (V (z))2 dz . 8 0

(7.4.25)

(7.4.26)

With the common separation of variables ψ(x, y, z) = Ψ (x, y)V (z) ,

and

V (z) + λ2 V (z) = 0 ,

(7.4.27)

the constraints (7.4.24) for ψ transform into ∆Ψ (x, y) − λ2 Ψ (x, y) = 0 , ∂ (Ψ + Bx ) = 0 , ∂n Ψ →0,

for (x, y) ∈ D+ , for (x, y) ∈ Γ , for x2 + y 2 → ∞ ,

(7.4.28)

where now, due to the periodic supports, λ = π/L. Moreover, with (7.4.25) expression (7.4.23) for K reduces considerably; all terms except the ﬁrst three are zero (or, better, O(δ 2 )). Thus, K for one rod becomes L λ2 x 2 ∂Bx 1 2 + (B + B )n V 2 (z)dz . (7.4.29) K=− (Ψ + Bx ) ds x y 2 Γ ∂n 2R2 x 0 In [122, Sect. 4], the details of the calculation of Bx , By and Ψ are presented, and as these analytical calculations are exact, this leads to an exact expression for the buckling current (exact within the concept of the slender beam theory). This eventually results in (see [122, eq. (5.17)]) 2 * πR E πR , (7.4.30) I0cr = √ L µ0 Q where

with β=

3 ∞ 1 − α2n 4 ) 2 Q = Q(d/R) = 2 nan , β n=1 1 + α2n +

d2 −1, R2

an =

α2n , (1 + α2n−2 )(1 + α2n+2 )

α=

(7.4.31)

d − 1 . (7.4.32) R

7.4 Superconducting Structures

L1

229

L2

t1

F t2 s1

R s2

I1 r1

I2 r2

0

Fig. 7.4. Two curves carrying current; generalization of the law of Biot and Savard √ Table 7.2. Values of Q and 1/ Q as functions of d/R d/R

1

1.5

2

3

4

6

8

10

Q√ 1/ Q

0.311 1.79

0.220 2.13

0.168 2.44

0.0935 3.27

0.0568 4.20

0.0266 6.13

0.0153 8.09

0.00985 10.08

√ Values for Q and 1/ Q as functions of d/R are given in Table √ 7.2. For later reference we note that for large values of d/R the factor 1/ Q approaches d/R. It is of technical interest to compare this exact result with the result of a less accurate, but much simpler solution procedure that is based upon a generalization of the law of Biot and Savart. The basic relation for this method is given by Moon in [156, eq. (2-6.4)]. Let L1 and L2 be two distinct curves in R3 , carrying the same electric current I0 , see Fig. 7.4. Moreover, let P1 and P2 be two points on L1 and L2 with position vectors r 1 (s1 ) and r 2 (s2 ), respectively, where s1 and s2 are the corresponding arc lengths. The force per unit of length in P1 acting on L1 is now calculated as the Lorentz force due to the current through L1 times the magnetic ﬁeld created by L2 . The latter follows from a generalization of the law of Biot and Savard as given in [156, eq. (2-6.3)–(2-6.4)]. According to [156, eq. (2-6.4)], this force is then given by (t1 (s1 ) × (t2 (s2 ) × R(s1 , s2 ))) µ0 I02 ds2 , (7.4.33) F (s1 ) = 4π R3 (s1 , s2 ) L2 where t1 and t2 are unit tangent vectors along L1 and L2 , respectively, and R is the position vector from P2 to P1 , i.e.

230

7 Magnetoelastic (In)stability and Vibrations

t1 =

dr 1 , ds1

t2 =

dr 2 , ds2

R = r1 − r2 .

(7.4.34)

The above formula for F is an approximation in so far as 1. the three-dimensional current-carrying bodies are considered as onedimensional curves (thus, for instance, the speciﬁc shape of the crosssection and the distribution of the current over the cross-section are disregarded); 2. the force due to the self-ﬁeld of L1 is neglected. Both these eﬀects are taken into account by the (exact) variational method. We shall show here that this Biot–Savard method will yield only good agreement with the exact results of the variational method if the two current ﬁlaments are suﬃciently apart from each other. To this end, we will apply the Biot–Savard method to the example above. In the symmetrical buckling mode the deﬂections of the two rods are equal but opposite and thus in the deﬂected state we have r 1 = (d + V (s1 )) ex + s1 ez , r 2 = (−d − V (s2 )) ex + s2 ez , R = (2d + V (s1 ) + V (s2 )) ex + (s1 − s2 ) ez , t1 = V (s1 ) ex + ez , t2 = −V (s2 ) ex + ez .

(7.4.35)

These formulas enable us to evaluate (7.4.33). In doing so, we must realize that the displacements are small and, hence, a linearization with respect to these displacements is allowed. In this sense, we approximate R3 by 6d R3 = R03 1 + 2 (V (s1 ) + V (s2 )) , (7.4.36) R0 where R0 = R0 (s1 , s2 ) =

4d2 + (s1 − s2 )2 ≥ 2d .

(7.4.37)

In the same way linearizing the numerator, we ﬁnd an expression for F of the form F (s1 ) = F (0) (s1 ) + f (s1 ) , (7.4.38) where F (0) is independent of, and f is linear, in the displacement V (s1 ). Hence, F (0) is the force in the pre-buckled state (causing the so-called prebuckling deﬂections), which may be discarded for the determination of the buckling current. Therefore, we deﬁne q(s1 ) as the force per unit of length in the x-direction acting on the deﬂected rod by q(s1 ) = (f (s1 ) · ex ) ,

(7.4.39)

and this force governs the deﬂection of the rod through the beam equation (from here on, we replace s1 by z, the axial coordinate of the rod)

7.4 Superconducting Structures

EIV iv (z) = q(z) .

231

(7.4.40)

The procedure described above yields the following expression for the load ∞ V (z) + V (ζ) − (z − ζ)V (ζ) 12d2 (V (z) + V (ζ)) µ0 I02 q(z) = − dζ . 4π R03 R05 −∞ (7.4.41) After two integrations by parts, in which it is used that V (z) is periodic in z, (7.4.41) becomes ∞ V (ζ) − V (z) µ0 I02 2 dζ . (7.4.42) V (z) + d q(z) = 4πd2 R03 −∞ At this step, we note that the second term on the right-hand side of (7.4.42) is of the order (d/L)2 with respect to the ﬁrst term, and since we want to restrict ourselves to cases where d L (in fact, d/L is small of the same order as δ), we may neglect this term. Thus, (7.4.40) becomes V iv (z) −

µ0 I02 V (z) = 0 . 4πd2 EI

(7.4.43)

For a simply supported rod, the boundary conditions are V (0) = V (L) = V (0) = V (L) = 0 ,

(7.4.44)

and the ﬁrst buckling mode that satisﬁes these boundary conditions is πz . (7.4.45) V (z) = A sin L After substitution into (7.4.43) this brings us to the following expression for the buckling current 2 * πR E I0cr = πd . (7.4.46) L µ0 √ Table 7.2 shows that 1/ Q ≈ d/R for values of d/R ≥ 4 (the relative difference is then less than 5%) and then (7.4.30) and (7.4.46) yield approximately the same result for I0cr . However, if the ﬁlaments are nearer to each other the correspondence becomes worse. In the limit d/R → 1, formula (7.4.46) predicts a buckling current that is about 80% lower than that obtained with (7.4.30). From this we conclude that the so-called direct Biot– Savard method yields acceptable results for the buckling current only if the current ﬁlaments are suﬃciently far away from each other. 7.4.2 A Set of Two Concentric Superconducting Rings Consider two concentric superconducting tori (or rings) both having a circular cross-section of radius R. The central line of the outer torus has radius

232

7 Magnetoelastic (In)stability and Vibrations b+d

b−d

2d

ez

R O

er D1

D2

Fig. 7.5. Cross-section of a pair of concentric tori

b + d, and that of the inner one b − d, where d > R. A system of cylindrical coordinates in the center O of the system is introduced as given in Fig. 7.5. We assume that ε = R/b 1, and that d/R is of order one; in view of this, the system of the two tori is called a slender system. The total current through each of the tori is called I0 , and the currents in the tori are equally directed. We consider only in-plane buckling, and we assume that the lowest buckling mode corresponds to an anti-symmetric buckling mode. The deﬂection of the outer torus (i = 1) and the inner (i = 2) torus can then be written as wi (φ)er + vi (φ)eφ , i = 1, 2, where w2 = −w1 for the anti-symmetric buckling mode. The tori are assumed inextensible, which can be shown to be expressible as vi (φ) + wi (φ) = 0 , i = 1, 2. In analogy with Bernoulli’s theory for the bending of slender beams, the 3-D displacement ﬁeld of the i-th torus may be written as, up to O(ε2 ), ur(i) = wi (φ) , (i)

uφ = vi (φ) −

(r − bi ) (wi (φ) − vi (φ)) , bi

i = 1, 2 ,

(7.4.47)

uz(i) = 0 , where b1 = b + d , b2 = b − d. The representation (7.4.47) yields the classical expression for the elastic energy of a slender inextensible ring in bending, 2π EIz (wi + wi )2 dφ (1 + O(ε)) , (7.4.48) Wi = 3 2b 0 where we have used that r = b(1 + O(ε)) , bi = b(1 + O(ε)) and where Iz = πR4 /4. As in the preceding examples, the perturbed potential ψ(r, φ, z) is separated according to

7.4 Superconducting Structures

ψ(r, φ, z) = Ψ (r, z)ω(φ) ,

233

(7.4.49)

where ω(φ) must be periodic in φ. For the ﬁrst buckling mode, we take ω(φ) = cos(2φ) and, accordingly, choose for the displacements of the central lines (7.4.50) w1 (φ) = w cos(2φ) , and w2 (φ) = −w cos(2φ) . This yields for the total elastic energy (7.4.48) of the system W = W1 + W2 =

9π 2 9π 2 ER4 2 2 EI w = w . z b3 4 b3

(7.4.51)

The electromagnetic interaction term K = I02 K is found in [218] in a fully analytical way, but we omit the details of this analysis here and refer to [218]. However, what we do want to point out is the following: for the slender system of tori we expect that on a local scale, i.e. for a small line element bdφ, the electromagnetic interaction between two such line elements of two diﬀerent tori will not diﬀer very much from the corresponding interaction between two parallel rods. This expectation is conﬁrmed by the result for K found in [218, eq. (4.84)]. In our notation, this result reads (K = K/I02 ) K=

µ0 bQ 2 w , 4R2

(7.4.52)

where Q = Q(d/R) is as given in (7.4.31). This yields for the buckling current * E 3πR3 . (7.4.53) I0 = 2 b µ0 Q Note that this result is of the same form as the resulting buckling current for two parallel rods as given in (7.4.30); the only diﬀerence is due to the elastic energy. The basic reason for this is that the expression for K as given by (7.4.29) for slender systems is dominated by its ﬁrst term. This term takes the same value for all types of slender systems that we will consider in this chapter, at least in a zeroth-order approximation with respect to ε. Since K is for one part determinant for the buckling value (the other part being determined by the elastic energy W ) we may state that the buckling current for any slender pair of parallel curved rods is equal to that of an equivalent pair of parallel straight rods times the ratio of the elastic energies. Of course, the concept “slenderness” has to be deﬁned properly in each problem at hand. As we did already in the example on the buckling of two parallel rods, we shall derive here also an approximate expression for the buckling current based on the so-called direct Biot–Savard method as given by Moon in [156, eq. (2-6.4)]. We will do this here not for the system of two concentric tori discussed above, but for a set of two parallel tori, or rings, as depicted in Fig. 7.6. The tori have equally directed currents. In this method, the tori are

234

7 Magnetoelastic (In)stability and Vibrations

b R

D1

ez d O

er d

R

D2 Fig. 7.6. Cross-section of a pair of parallel tori

considered as 1-dimensional rings. We consider here out-of-plane buckling, which consists of out-of-plane bending (wi (φi )) and torsion (τi (φi ): the twist per unit of length along the central line). The positions on the two deﬂected rings (i = 1, 2) are then given by r 1 = b er1 (φ1 ) + (d + w1 (φ1 )) ez , r 2 = b er2 (φ2 ) + (−d + w2 (φ2 )) ez ,

(7.4.54)

where ri , φi , z are cylindrical coordinates connected to the i-th ring. In a way analogous to that used in the derivation of (7.4.43) we can also here derive an expression for the force per unit of length on one ring due to the current in the other ring. In this case, this force is in the z-direction and equal to, for the ﬁrst ring and when neglecting O(d/b)-terms, fz (φ1 ) =

µ0 I02 [w1 (φ1 ) − w2 (φ1 )] . 8πd2

(7.4.55)

The ring equations for out-of-plane bending and torsion can a.o. be found in [156, eq. 6-7.18]; with A = EI = (π/4)ER4 and C = GIp = (π/2)ER4 /(1+ν) and in our notations they read −

EI GIp [w1iv (φ1 ) − bτ1 (φ1 )] + 4 [w1 (φ1 ) + bτ1 (φ1 )] + fz (φ1 ) = 0 , b4 b EI GIp − 2 [w1 (φ1 ) − bτ1 (φ1 )] − 2 [w1 (φ1 ) + bτ1 (φ1 )] = 0 . b b (7.4.56)

Putting wi (φi ) = Wi cos(2φi ) ,

τi (φi ) = Ti cos(2φi ) ,

i = 1, 2 ,

(7.4.57)

7.4 Superconducting Structures

235

and using GIp = EI/(1 + ν), we obtain from the second relation of (7.4.56) for i = 1 4(2 + ν) W1 . (7.4.58) T1 = − (5 + ν) b With this result the ﬁrst equation of (7.4.56) yields 36EI µ0 I02 W1 cos(2φ) = fz (φ) = [W1 − W2 ] cos(2φ) . 4 (5 + ν)b 8πd2

(7.4.59)

An analogous relation holds for W2 . It is then easily seen that the lowest buckling current occurs for W2 = −W1 (anti-symmetric buckling mode) and is equal to (with I = πR4 /4) * E 6πdR2 . (7.4.60) I0 = b2 µ0 (5 + ν) As can be found in [218], the “exact” variational method would yield for the buckling value * E 6πR3 I0 = . (7.4.61) b2 µ0 (5 + ν)Q √ Hence, the result (7.4.60) is only in agreement with (7.4.61), if 1/ Q = d/R, a result that was already found at the end of Sect. 7.4.1. Thus, we conclude again that the direct Biot–Savard method yields an acceptable approximation (relative error less than 5%) for the buckling current only if the current ﬁlaments are far enough away from each other, say d/R ≥ 4. It seems superﬂuous to say that an analogous result can also be obtained for the system of two concentric tori; see [218]. 7.4.3 How to Use the Law of Biot and Savard in the Variational Principle In the examples presented in the preceding sections, we used the variational principle in the exact sense. This means that we looked for and found exact rigid-body ﬁelds B 0 and perturbed ﬁelds b, which we used in the variational expression for K, thus obtaining exact values for the buckling current. Exact always means here within the range of slender-beam theory, so for R/l 1, where R is the radius of the circular cross-section of the beam and l a characteristic measure of length of the beam. Essential in our model for a superconductor is the assumption that the current runs over the surface of the superconductor only, thus shielding the body from a magnetic ﬁeld. Hence, there is no magnetic ﬁeld inside the body. In the preceding section, we also presented a more direct method based on a formula for the Lorentz force on a current carrier due to the current in another conductor derived from the Biot–Savard law. We referred to this method as the direct Biot–Savard

236

7 Magnetoelastic (In)stability and Vibrations

method. The second method is less exact than the variational method, but much easier to work with in practice. A comparison of the two methods showed reasonable agreement as long as the conductors are not too close to one another. In this respect, it seems reasonable to look for a method that combines the advantages of the two methods, i.e. the greater exactness of the variational method and the convenience in the use pertinent to the direct Biot–Savard method. To this end, we shall use in the variational formulation for K an admissible magnetic ﬁeld b obtained on the basis of the law of Biot and Savard. Our expectation that this combined approach will yield a useful approximation for the buckling current is supported by the observed correspondence between the results of the two respective methods. As this method consists of a combination of the variational principle and the Biot–Savard law, we will refer to this method as the combined method. Our hope that this method will result in buckling values that are closer to the exact values will be conﬁrmed by the results to be presented in the next section. Moreover, we will show there how we can obtain in a relatively easy way buckling values for systems that we can not solve in an exact way. In the next section, we shall present the detailed derivation of this method; moreover, we shall give the speciﬁc nature of the simpliﬁcations and the restrictions under which they are allowable. Applications to such complex systems as sets of helical or spiral conductors will also be presented. The combined method will calculate in a convenient way buckling currents for more or less complex structures, which are suﬃciently precise in practical applications.

7.5 Some Results for Superconducting Structures 7.5.1 Review of Speciﬁc Structures and Some Results In this section, we present a series of examples of applications of the variational principle described in the preceding chapter to superconducting systems. These systems always consist of slender rod-like structures, of circular cross-section, radius R, and carrying a total current I0 . These examples concern: 1. A set of two parallel rods, inﬁnitely long, but periodically supported over distances L, and a distance 2d apart from each other (R < d L); cf. [122]; see also Sect. 7.4.1. 2. A set of two concentric tori (or rings) in one plane; the rings have radii b1 and b2 , and the distance between them is 2d = b2 − b1 (R < d b2 ); cf. [218]; see also Sect. 7.4.1. 3. A set of two identical parallel tori (or rings), of radius b and distance 2d (R < d b); cf. [218].

7.5 Some Results for Superconducting Structures

237

4. Sets of n , n > 2, equidistant parallel rods (like the set in example 1.); cf. [124]; see also Sect. 7.5.2. 5. An inﬁnite helical conductor, periodically supported at every n turns; the radius of the helix is b and the pitch is h, the distance between two turns is 2d = 2πh, and the support length is 2πbn (R < πh πb), cf. [256] or [257]; see also Sect. 7.5.3. 6. A ﬁnite helical conductor (like the one in example 5.) of n turns, simply supported at its end points; cf. [257]; see also Sect. 7.5.3. 7. A ﬂat spiral of n turns; the radius is given as a function of the arc ϕ, for ϕ ∈ [0, 2πn], by b(ϕ) = b0 + hϕ, where h is the constant pitch; the distance between two turns is 2d = 2πh (R < d b0 ); the spiral is simply supported at its end points; cf. [257]; see also Sect. 7.5.3. Problems 1, 2, and 3 are solved completely by analytical means, using conformal mapping and complex function theory. These mathematical procedures were necessary to obtain an exact solution for the magnetic ﬁelds, both for the rigid-body and the perturbed states (referring to the deformed state of the structure), which, in turn, were needed for the calculation of the electromagnetic interaction integral K, as given in (7.4.23). In this way, an exact value for the buckling current is obtained (exact within the concept of slender-beam theory, i.e. up to O(R2 /L2 )-terms). The analytical approach to problem 4 resulted in a set of integral equations for the perturbed magnetic ﬁeld, which had to be solved numerically. Nevertheless, the buckling value found was exact, in the same sense as in the preceding examples. For problems 5 to 7 no analytical solution was found, but here the variational principle was used to ﬁnd a good approximation for the buckling current; for this approximation the law of Biot and Savard was used to obtain an admissible magnetic ﬁeld for the variational method; this method was already explained in Sect. 7.4.3. In the latter three problems a result is also used that states that the value of K for a set of two (or more) rings is equal to the value of K for a set of two straight parallel rods. This result was proved to be true for the set of two concentric rods; see also Sect. 7.4.2. It can also be made plausible by the following reasoning: consider two rings and take a point A1 on one of the rings; then the interaction of the second ring with A1 is concentrated at a point A2 and its direct neighbourhood, where A2 is that point of the other ring that is closest to A1 . Then, according to the slenderness condition, L d, in a d-environment of A2 the ring may be considered to be locally straight. The errors induced in this way are of O(d2 /L2 ). This reasoning also applies to a slender helix or spiral and, therefore, the contribution to K of two interacting turns of a helical or spiral coil can be calculated by replacing them locally by a set of two parallel rods, for which the value of K is known. Results for the buckling current for the 7 examples listed above are presented in Table 7.3. We conclude that for all the problems considered here the buckling current I0cr contains the common factor J , where J = (dR2 /L2 ) E/µ0 , while the coeﬃcient preceding J , is only a function of

238

7 Magnetoelastic (In)stability and Vibrations

Table 7.3. Buckling currents I0cr for problems 1 to 7; E and ν areYoung’s and Poisson’s moduli, respectively, the common factor J = (dR2 /L2 ) E/µ0 , where L = πb for problems 2, 5, and √ 6, L = π(b1 +b2 )/2 for problem 3, and L = π(b0 +nπh) for problem 7, and q = R/d Q, with Q = Q(d/R) according to (7.4.31); the factors αn = αn (n) are given below, and κ = κ(d/R) , N (n) , λ(n), and λ(n, h) will be speciﬁed in the next sections System I0cr

1.

2.

π 3 qJ

3π 3 qJ

System I0cr

3. 6π 3 q √ 5+ν

5. π2

-

4. J

αn π 3 qJ

6.

N (n) (2n−1)(1+ν)κ

J

π3

-

2λ(n) (1+ν)κ

7. J

π3

-

2λ(h,n) κ

J

R/d and is diﬀerent for each problem. The factor αn (n) is for n = 3, 4, 5, and ∞ given by α3 = 2/3 , α4 = 0.753 , α5 = 0.723, and α∞ = 2/π = 0.637. Formulas for κ = κ(d/R) , N (n) , λ(n), and λ(n, h) will be derived in the subsequent sections. In the next section, we shall show how we can use the law of Biot and Savard in the variational method. The basic idea behind this approach is that the magnetic ﬁeld obtained by the law of Biot and Savard satisﬁes the constraints of the variational principle and, hence, constitutes an admissible ﬁeld. In the same way as shown at the end of Sect. 7.4.1, we can directly from the law of Biot and Savard calculate the Lorentz forces on the structures and thus ﬁnd approximate values for the buckling currents. These values can be obtained from Table 7.3 by substituting q = 1 for problems 1 to 4, and κ = 1 for problems 5 to 7. What we will ﬁnd with the combined Biot– Savard-variational method are buckling values in between the values from the variational method (exact) and the direct Biot–Savard method. Hence, the proposed combined method yields, on the one hand, an improvement of the direct Biot–Savard method, and, on the other hand, an easy way to obtain an admissible magnetic ﬁeld. The details of the derivation of the combined method will now be presented in the following section. 7.5.2 The Combined (Variational Biot-Savard) Method Before starting with the derivation of the combined method, we ﬁrst shortly recapitulate from Sect. 7.4.1 the main lines of the variational principle for a superconducting body in vacuum. In this case, there is only a magnetic ﬁeld `re’s law B in the vacuum surrounding the body, which has to satisfy Ampe (see (7.4.2)) (B, ds) = µ0 I0 , C

(7.5.1)

7.5 Some Results for Superconducting Structures

239

where C is a contour entirely in the vacuum, encircling the current carrier and I0 is the prescribed current through the superconductor. The variational method is based upon a chosen expression for the Lagrangean for an elastic superconducting body, which, according to (7.4.1), is given by 1 (B, B)dV − ρU dV , (7.5.2) L= 2µ0 G+ G− where the ﬁrst integral represents the magnetic energy of the vacuum ﬁeld and the second is the elastic energy of the deformed body. Note that here G+ and G− are the conﬁgurations of the vacuum and the body, respectively, in the deformed state. Variation of L should satisfy the constraint (7.5.1) and eijk Bk,j = 0 ,

x ∈ G+ ,

and

B→0,

|x| → ∞ .

(7.5.3)

The total ﬁeld B is split into the rigid-body ﬁeld B 0 and the perturbed ﬁeld b, where the latter is due to the displacement u of the elastic body. The perturbed ﬁeld is expressed in the perturbed magnetic potential ψ by bi = ψ,i ,

(7.5.4)

where this potential has to satisfy the extra constraint (see (7.4.18)) ψ,ii = 0 ,

and

ψ(x) → 0 ,

|x| → ∞ .

(7.5.5)

The second variation of L, called J, can then be determined in terms of B 0 , ψ, u and the associated linear deformations, or strains, eij = (ui,j + uj,i )/2, eventually resulting in the expression (7.4.19). In the examples presented thus far, we used the exact solutions for B 0 (x) and b(x). However, if the systems become more complex, it becomes increasingly diﬃcult, not to say impossible, to determine these ﬁelds exactly. Therefore, we are looking for admissible ﬁelds from which we may hope that they are not too far away from the exact ones. Admissible ﬁelds are ﬁelds B 0 (x) and ψ(x) that satisfy the constraints (7.5.1), (7.5.3) and (7.5.5). We will employ these ﬁelds to obtain an approximation for the functional J. Here, we will use the Biot–Savard ﬁelds as they can be derived from the law of Biot and Savard (see [156, eq. (2-6.3)]) (t(s) × R(x, s)) µ0 I0 ds , (7.5.6) B(x) = 4π R3 (x, s) L where t(s) is the unit tangent vector along L in a point P on L having arc length s, and R(s, x) is the position vector of the point x ∈ G+ with respect to P, while R = |R|. We will show that these ﬁelds are admissible. The already rather good correspondence between the results of the direct Biot–Savard method and the variational method as found in the earlier

240

7 Magnetoelastic (In)stability and Vibrations

sections, supports us in our opinion that this choice will lead us to a good approximation of the buckling current. Before proceeding with the explicit derivation of the admissible ﬁelds, we ﬁrst simplify expression (7.4.19). To this end, we realize that we wish to apply the variational principle to slender, beam-like, bodies. In the bending of such structures the pure deformations are much smaller than the local rotations of the slender body. This motivates us to neglect in the magnetic term in (7.4.19) those terms that contain a factor of order B 2 (here, the small parameter is some norm of the deformations eij ). Moreover, since the term containing the prestresses Tij in the elastic energy integral is small of the same order, this term must be neglected too. We like to mention that the neglect of the above terms leads to an error that is small of order R/l and, hence, is justiﬁed for systems of slender bodies. Altogether, this leads us to the following simpliﬁed expression for J (in which B is used for B 0 ): ν E ekk ell + ekl ekl dV J =− 2(1 + ν) G 1 − 2ν 1 [(ψ + Bk uk ),j Bj ui + (ψ,i + Bi,j uj − Bj ui,j )ψ]Ni dS − 2µ0 ∂G = −W + I02 K .

(7.5.7)

We have written the last term in (7.5.7) as I02 K, because K is independent of I0 then. This is true because B and ψ are both linear in I0 . With Bi,j = Bj,i , as follows from (7.5.3), we can reduce the integral I02 K still somewhat further to obtain 1 2 [(ψ + Bk uk ),j (Bj ui Ni + ψNj )]dS , (7.5.8) I0 K = − 2µ0 ∂G where we have neglected again a term proportional to B 2 . It is possible to derive explicit expressions for admissible B(x) and ψ(x) from (7.5.6) for arbitrarily curved circuits, but we refrain from doing so here. Instead, we use here a result of Sect. 7.5.1 that taught us that for the derivation of the K-integral it suﬃces to calculate its value for a corresponding system of two parallel rods. We generalize this result by stating that this value for K can also be used for systems of curved beams, not only for a pair of rings (as we showed in Sect. 7.4.2), but also for such structures as helical or spiral conductors, provided the system satisﬁes the condition of being slender. Hence, in this case we only need to calculate B(x) and ψ(x) for a straight current carrier L as we shall ﬁrst do now. Consider an inﬁnite one-dimensional conductor L , carrying a current I0 . Let L in its original state be given by a straight line along the e3 -axis, while a point P on L then is given by its, undeformed, position vector ξ = ζe3 . The rigid-body ﬁeld B(x) for x ∈ G+ , with x = xe1 + ye2 + ze3 , follows then from (7.5.6) as

7.5 Some Results for Superconducting Structures

B(x) =

µ0 I0 4π

∞

−∞

241

(e3 × (x − ξ)) dζ |x − ξ|3/2

∞

=−

µ0 I0 (ye1 − xe2 ) 4π

=−

µ0 I0 (ye1 − xe2 ) . 2π (x2 + y 2 )

−∞

[x2

+

y2

dζ + (z − ζ)2 ]3/2 (7.5.9)

Clearly, this ﬁeld satisﬁes the constraints (7.5.1) and (7.5.3). For the calculation of ψ(x) we have to consider the deﬂected beam. For this, we assume that L has a deﬂection in the e1 -direction: u = u(ζ)e1 , by which t(ζ)

= u (ζ)e1 + e3 ,

(7.5.10) R(x, ζ) = (x − u(ζ))e1 + ye2 + (z − ζ)e3 , 3x R−3 = R0−3 1 + 2 u(ζ) , R0 = [x2 + y 2 + (z − ζ)2 ]1/2 , R0 where the last result is only correct up to ﬁrst order terms in u. With this, (7.5.6) yields for the perturbed ﬁeld in x ∈ G+ ∞ 1 3x µ0 I0 b(x) = 3 2 (xe2 − ye1 )u(ζ) 4π −∞ R0 R0 −[(z − ζ)e2 − ye3 ]u (ζ) − u(ζ)e2 dζ ∞ u(ζ) 3xy 5 dζ e1 −∞ R0 1 3x2 + (7.5.11) u(ζ) + (z − ζ)u (ζ) − 2 u(ζ) dζ e2 3 R0 −∞ R0 ∞ u (ζ) −y dζ e 3 R03 −∞ ∞ 1 3xy µ0 I0 3y 2 3y(z − ζ) =− e − − + e e u(ζ)dζ , 1 2 3 4π R05 R03 R05 R05 −∞

=−

µ0 I0 4π ∞

where the latter step follows after an integration by parts. From (7.5.6) it follows that eijk bk,j = 0, and then there exists a magnetic potential ψ(x) such that (7.5.12) bj = ψ,j . This potential is equal to µ0 I0 ψ(x) = 4π

∞

−∞

yu(ζ) dζ . R03

(7.5.13)

242

7 Magnetoelastic (In)stability and Vibrations

It is easily checked that (7.5.13) implies ψ,ii = 0 and ψ(x) → 0 for |x| → ∞, so this result indeed satisﬁes the constraints (7.5.5). We assume that the current carrier L is periodically supported over distances l. In that case, u(z) = 0 in all points z = kl , k ∈ N, and, moreover, u(z) is periodic in z with period 2l. We thus can write (7.5.13) as ∞ ∞ 1 u(ζ) − u(z) µ0 I0 y u(z) ψ(x) = dζ + dζ 3 4π R03 −∞ R0 −∞ 2 µ0 I0 2yu(z) R R =− ln 1+O . (7.5.14) 4π (x2 + y 2 ) l2 l In the latter step we have used the mean value theorem and have assumed that for x → ∂G, x and y are such that (x2 + y 2 )/l2 = O(R2 /l2 ); in all our applications the latter condition is fulﬁlled. In the next sections, we shall employ the results derived above for the solution of the buckling problem for helical and spiral superconductors. However, before doing so, we ﬁrst apply the method to our standard example of a set of two parallel straight rods. We do this for three reasons: 1. as an illustration of the method presented in this section; 2. because, and this is the most important reason, one of the results of this example, namely the value of the integral K, can directly be applied to problems more complex than those that will be dealt with in the subsequent sections; 3. in order to show that the combined method leads us to a result that is closer to the (exact) result of the variational method than the direct Biot–Savard method does.

Example: A set of two parallel superconducting rods (combined method) We consider here exactly the same problem as in the Example of Sect. 7.4.1. Thus two parallel rods with equally directed currents of magnitude I0 , and bending in the e1 , e3 -plane. The distance between the rods is 2d, the rods are periodically supported over distances L , (l = L), and the ﬁrst buckling mode is the anti-symmetric buckling mode. This brings us to the following choice for the displacements of the two rods: πz e1 , L πz e1 . u(2) (z) = −V (z)e1 = −A sin L u(1) (z) = V (z)e1 = A sin

(7.5.15)

From (7.5.9) and (7.5.14) we ﬁnd the ﬁelds B(x) and ψ(x) in an arbitrary point x in the vacuum by simple superposition. This results in

7.5 Some Results for Superconducting Structures

B(x) = −

µ0 I0 2π

y y + 2 r12 r2

and ψ(x) =

µ0 I0 2π

e1 −

x+a x−a + r12 r22

243

e2

y y πz , − A sin r12 r22 L

(7.5.16)

(7.5.17)

where r12 = (x + a)2 + y 2 ,

r22 = (x − a)2 + y 2 .

(7.5.18)

for the two rods and over one full period −L < We can then calculate z < L from (7.5.8) by substituting (7.5.16) and (7.5.17) into this expression. After some lengthy but elementary calculations this results in L µ0 I02 κ µ0 I02 κL 2 I02 K = [u(1) (z) − u(2) (z)]2 dz = A , (7.5.19) 2 16πd 4d2 −L I02 K

where κ=

1 − 4(R/2d)2 + 32 (R/2d)4 . [1 − (R/2d)2 ]2

(7.5.20)

The elastic energy for the two rods over one full period is, like in (7.4.26), W =

π ER4 4

L

−L

(V (z))2 dz =

π6 ER4 . 4L3

By combining (7.5.19) and (7.4.32), we obtain the buckling value * + W E π 3 dR2 = . I0cr = 2 K L µ0 κ

(7.5.21)

(7.5.22)

We have compared this value with the corresponding results of the Example of Sect. 7.4.1 for the variational method and the direct Biot–Savard method; the result is depicted in Fig. 7.7. In this graph the buckling currents according to the variational method (VM) and to the combined method (CM), both normalized with respect to the buckling current from the direct Biot–Savard method (BS), are displayed as function of d/R. The graph shows that the CM-value lies above the BS-value and below the VM-value. Hence, as could be expected, the combined method yields an improvement of the BS-value in the direction of the VM-value. The diﬀerences are significant, but of technical relevance only for values of d/R < 4. Moreover, the CM-value is a conservative estimate, meaning that the “real” buckling value is higher. So, using the CM-estimate keeps the designer on the safe side. The reason for the higher VM-value is the following: In the direct BS-method the currents run through the 1-dimensional central lines of the rods, whereas in the VM-approach the current is distributed over the boundary of the crosssection. When the currents in the two rods run in the same direction, the interaction between the currents causes a shift of the current distributions

244

7 Magnetoelastic (In)stability and Vibrations I0cr (BS)

I0cr

VM

1.4 +

1.3 +

1.2 +

CM

1.1 +

1+

BS

|

|

0

1

|

|

2

3

|

4

|

5

|

6

d/R

Fig. 7.7. Comparison of the buckling currents for a set of two parallel rods according to the variational method (VM) and the combined method (CM), both normalized with respect to the buckling current from the direct Biot–Savard method (BS); from [257]

to the most opposite sides. If we then would try to imagine that the current runs through a 1-dimensional line, this line would be shifted away from the central line to the outer side. Thus, apparently the distance between the rods has increased and then the set buckles for a higher current. This explains why the VM-value is higher than the CM- or BS-value. We can easily generalize the above method to sets of n , n > 2, parallel rods. Realizing that the integral in (7.5.19) is built up of two integrals (one for the ﬁrst rod with respect to the second one and vice versa), we see that the generalization to n rods is straightforward, and yields 1 µ0 I02 κ ) ) = 32πd2 i=1 j=1 (j − i)2 n

I02 K

n

L

−L

[u(i) (z) − u(j) (z)]2 dz .

(7.5.23)

j =i

This formula can also be applied to more complex structures such as helical or spiral conductors. Then, n denotes the number of turns of the structure and 2d is the distance between two subsequent turns. These cases will be discussed in the following sections.

7.5 Some Results for Superconducting Structures

245

7.5.3 Helical or Spiral Superconductors Consider a superconductor in the form of a cylindrical helix; see Fig. 7.8. The radius of the helix is b and the constant pitch is h. For the distance between two turns, we then have 2d = 2πh; for a slender helix πh b. In the undeformed conﬁguration a point on the central line of the helix is given by its position vector X = X(ϕ) = b cos ϕ ex + b sin ϕ ey + hϕ ez = b er (ϕ) + h ϕez . (7.5.24) In this section, we shall consider both ﬁnite helices of n turns (in this case ϕ ∈ [0, 2πn]) and inﬁnite ones (ϕ ∈ R). In the ﬁrst case, the helix is simply supported in its end points, whereas in the second case the helix is periodically supported in the points ϕ = ϕk = 2kπn , k ∈ N. The total current through the helix is I0 . Since the pitch angle α is very small (α ≈ b/h 1) the dominant buckling mode will be a displacement in the axial or z-direction, i.e. u = u(ϕ)ez .

(7.5.25)

Here, u(ϕ) is the displacement of the central line of the helical conductor, causing, besides bending, also torsion of the helix. We denote the torsion angle by β(ϕ). This torsion has no inﬂuence on the integral K; it only aﬀects the elastic energy W . By variation of W with respect to β we will derive a relation expressing β(ϕ) in terms of u(ϕ). For the determination of the buckling current, we need expressions for the integrals K and W . The basic idea for K is completely analogous to that leading to (7.5.23). We only have to replace d by πh to obtain n n 1 µ0 κ ) ) [u(i) (ϕ) − u(j) (ϕ)]2 bdϕ , K= 32π 3 h2 i=1 j=1 (j − i)2 Li

(7.5.26)

j =i

I0 b ez ϕ ex 2πh

Fig. 7.8. The helical conductor

246

7 Magnetoelastic (In)stability and Vibrations

where Li stands for the trajectory Li = {ϕ | 2(i − 1)π ≤ ϕ ≤ 2iπ} ,

(7.5.27)

while, for ϕ ∈ Li u(i) (ϕ) = u(ϕ) ,

and

u(j) (ϕ) = u(ϕ + 2(j − i)π) .

(7.5.28)

For the elastic energy, we use the classical expression for a slender curved rod GIp EI (u − bβ)2 b dϕ + 4 (u + bβ )2 b dϕ W = 4 2b 2b L L 1 πER4 2 2 (u + bβ ) dϕ , (7.5.29) = (u − bβ) + 8b3 (1 + ν) L where L stands for the total length of the helix: L = L1 ∪ L2 ∪ ....Ln . In the following two subsections, we ﬁrst consider the inﬁnite and after that the ﬁnite helix. The Inﬁnite Helix For the inﬁnite helix, periodically supported over n turns, we choose for the lowest buckling mode ϕ . (7.5.30) u(ϕ) = A sin 2n Due to the periodicity, we may restrict the integral for K to that for one full period, say ϕ ∈ [0, 4πn). According to (7.5.26) we then obtain K=

2n ∞ µ0 κb ) ) 1 [u(ϕ) − u(ϕ + 2(j − i)π)]2 dϕ 32π 3 h2 i=1 j=−∞ (j − i)2 Li j =i

2 4πn " ∞ ϕ πk ϕ µ0 κb ) A2 − sin + sin = 32π 3 h2 k2 0 2n 2n n k=1 # 2 ϕ πk ϕ − dϕ + sin − sin 2n 2n n ∞ πk µ0 κbn ) A2 1 µ0 κb = 1 − cos 1− = A2 . (7.5.31) 4π 2 h2 k2 n 8h2 2n k=1

In accordance with (7.5.30) we assume the torsion β to have the form ϕ 1 β(ϕ) = B sin . (7.5.32) b 2n Substituting (7.5.30) and (7.5.32) into (7.5.29), we obtain for the elastic energy of one full period

7.5 Some Results for Superconducting Structures

π 2 ER4 W = 8b3

4πn

0

A +B 4n2

2 sin2

247

ϕ 2n

! ϕ 1 2 2 (A + B) cos + dϕ 4(1 + ν)n2 2n

π 2 nER4 = 8b3

A +B 4n2

2

1 + (A + B)2 4(1 + ν)n2

! . (7.5.33)

Since the variation of J with respect to β must be zero and K is independent of β, or B, we have ∂W/∂B = 0, yielding B=−

(2 + ν) A, 4(1 + ν)n2 + 1

(7.5.34)

and with this W =

π 2 nER4 π 2 ER4 A2 (4n2 − 1)2 A2 = [1+O(n−2 )] , (7.5.35) 3 4 2 4b 16n [4(1 + ν)n + 1] 16(1 + ν)nb3

for n 1. With use of (7.5.31) and (7.5.35) in J = −W + I02 K, we ﬁnd from J = 0 the following expression for the buckling current * E N (n) πhR2 , (7.5.36) I0cr = 2 b (2n − 1)κ (1 + ν)µ0 where κ is given by (7.5.20) and N (n) by N (n) =

' 1− 1+

(

1 2 4n2 1 2 4(1+ν)n

= 1 + O(n−2 ) ,

(7.5.37)

for n 1. According to its deﬁnition, N (n) depends on ν but only in a very weak sense; for ν running from 0 to 0.5, the value of N (n) changes by less than 2%. The Finite Helix Consider a ﬁnite helix of n turns, of the same type as described in the preceding section, thus with ϕ running from ϕ = 0 to 2πn. For the evaluation of K and W we have to select a representation for the deﬂection u(ϕ) and the associated torsion β(ϕ). Due to the ﬁniteness of the helix this representation

248

7 Magnetoelastic (In)stability and Vibrations

is no longer as simple as it was for the inﬁnite helix. We shall assume a representation in a series of N sine-functions2 for u(ϕ) according to (we assume the ﬁrst buckling mode to be symmetric about ϕ = πn) u(ϕ) =

N )

Ak sin

k=1

(2k − 1)ϕ . 2n

(7.5.38)

We choose for the torsion the analogous representation β(ϕ) =

N ) k=1

(2k − 1)ϕ Bk sin . 2n

(7.5.39)

Substituting (7.5.38) and (7.5.39) into expression (7.5.29) for W , we arrive at " #2 4πn ) 2k − 1 2 (2k − 1)ϕ πER4  (Ak + Bk ) cos W = 8(1 + ν)b3 0 2n 2n k

" +(1 + ν)

) (2k − 1)2 4n2

k

π 2 nER4 ) = 8b3

"

k

Ak + Bk

(2k − 1)ϕ sin 2n

 #2  dϕ (7.5.40)

2 # (2k − 1)2 (2k − 1)2 2 (Ak + Bk ) + Ak + Bk . 4(1 + ν)n2 4n2

From ∂W/∂Bk = 0 we obtain Bk = −

(2 + ν)(2k − 1)2 Ak , [4(1 + ν)n2 + (2k − 1)2 ]

(7.5.41)

yielding, after substitution into (7.5.40), W =

n ) πER4 ωk A2k , 8(1 + ν)b3

(7.5.42)

k=1

with π (2k − 1)2 ωk = 4n 2

(2 + ν)2 (2k − 1)2 (1 + ν)(2k − 1)2 − 1+ . 4n2 [4(1 + ν)n2 + (2k − 1)2 ] (7.5.43)

In [257], also a representation in splines was given. However, both the analytical and the subsequent numerical analysis were much more cumbersome and time-consuming than the sine-representation, while the ultimate results were practically the same. Therefore, we present here only the results of the sinerepresentation.

7.5 Some Results for Superconducting Structures

249

From (7.5.42) we conclude that in the elastic energy W the Ak -modes are uncoupled. For the magnetic integral K we obtain from (7.5.26) (with m = j − i)

) n n−i ) (2k − 1)ϕ 1 µ0 κb ) Ak sin K= 32π 3 h2 i=1 m2 Li 2n k

m=−(i−1)

m =0

(2k − 1)ϕ (2k − 1)πm + − sin 2n n

2 dϕ

2π(n−m) ) n−1 (2k − 1)ϕ µ0 κb ) 1 A sin = k 16π 3 h2 m=1 m2 0 2n k

(2k − 1)ϕ (2k − 1)πm + − sin 2n n =

2 dϕ

N µ0 κb ) kkl Ak Al , 16π 3 h2

(7.5.44)

k,l=1

with kkl =

(2k − 1)πm π 1 2n (n − m) − sin m2 n (2k − 1) n m=1 (2k − 1)πm × sin2 , 2n if k = l ; n−1 )

n−1 )

(7.5.45)

(k + l − 1)πm 1 sin (k + l − 1) n m=1 (k − l)πm (2k − 1)πm (2l − 1)πm 1 sin + sin sin , (k − l) n 2n 2n if k = l .

kkl = −

2n m2

With (7.5.42) and (7.5.44) we obtain for J = −W + I02 K,   N N ) ) πER4  ωk A2k − λ kkl Ak Al  , J =− 8(1 + ν)b3 k=1

where λ=

(7.5.46)

k,l=1

κb4 (1 + ν)µ0 2 I0 . 2π 4 h2 R4 E

(7.5.47)

250

7 Magnetoelastic (In)stability and Vibrations

Variation of J with respect to Ak yields the linear eigenvalue problem ωk Ak − λ

N )

kkl Al = 0 ,

k = 1, 2, . . . , N .

(7.5.48)

l=1

This problem is solved numerically, and from the lowest eigenvalue λ the buckling current is calculated. This eigenvalue is independent of all system parameters except for the number of turns n (and also, but only in a very weak sense, for ν). Hence λ = λ(n), and from (7.5.47) it then follows that * * 2π 2 λ(n) πhR2 E I0cr = . (7.5.49) 2 (1 + ν)κ b µ0

The Flat Spiral As a ﬁnal example, consider a ﬂat spiral conductor, lying in the xy-plane; see Fig. 7.9. In the undeformed conﬁguration, a point on the central line of the spiral is given by its position vector X = X(ϕ) = b(ϕ) cos ϕ ex + b(ϕ) sin ϕ ey = b(ϕ)er (ϕ) , with

2πh

ey

er ϕ

ex

I0 b

Fig. 7.9. The ﬂat spiral conductor

(7.5.50)

7.5 Some Results for Superconducting Structures

b(ϕ) = b0 + hϕ ,

0 ≤ ϕ ≤ 2πn ,

for

251

(7.5.51)

where b0 is the radius of the spiral at its starting point, 2πh is the distance between two adjacent turns (d = πh), and n is the number of turns. The cross-section of the conductor is circular, of radius R. The system is called slender if R/b0 1 (then, R/b(ϕ) < πh/b(ϕ) 1 for all ϕ ∈ [0, 2πn]). The spiral is simply supported in the two end points. We assume that in the dominant buckling mode the spiral deforms in its plane and that the pertinent displacement of its central line is given by (in polar coordinates) (7.5.52) u = u(ϕ) = u(ϕ)er + v(ϕ)eϕ . The spiral is taken to be inextensible, implying that u(ϕ) + v (ϕ) = 0 .

(7.5.53)

For an inextensible slender curved (almost circular) beam the elastic energy due to in-plane bending is given by 2πn 1 π 4 [u (ϕ) + u(ϕ)]2 dϕ . W = ER (7.5.54) 3 (ϕ) 8 b 0 Completely analogous to the preceding sections, the expression of K follows again from (7.5.26); we must only realize that now b = b(ϕ). Just as in the preceding section we use for u(ϕ) a series of N sinefunctions; see (7.5.38). Using this representation in W , we arrive at π W = ER4 8 =

πER4 8b30

2πn

0

k

N ) k,l=1

where

! 2 ) kϕ 1 k2 dϕ 1 − 2 Ak sin b3 (ϕ) 4n 2n

Ikl = 0

k2 l2 1− 2 1 − 2 Ikl Ak Al , 4n 4n

2πn

1 sin β 3 (ϕ)

and β(ϕ) =

kϕ 2n

sin

h b(ϕ) =1+ ϕ. b0 b0

lϕ dϕ , 2n

(7.5.55)

(7.5.56)

(7.5.57)

The evaluation of K is analogous to that in the preceding section and leads to

252

7 Magnetoelastic (In)stability and Vibrations

2π(n−m) n−1 µ0 κb0 ) 1 K= β(ϕ + πm) 16π 3 h2 m=1 m2 0 ×

" ) k

=

#2 kϕ kϕ kπm + Ak sin dϕ − sin 2n 2n n

(7.5.58)

n−1 N kπm lπm µ0 κb0 ) 1 ) J A A sin sin , klm k l 16π 3 h2 m=1 m2 2n 2n k,l=1

where

2π(n−m)

Jklm = 4

β(ϕ) cos πm

kϕ 2n

cos

lϕ dϕ . 2n

(7.5.59)

Introducing λ as (we use here b1 = b0 + nπh, the mean radius of the spiral, instead of b0 in order to weaken the inﬂuence of h on λ) λ=

µ0 κb41 I2 , 2π 4 ER4 h2 0

(7.5.60)

substituting (7.5.55) and (7.5.58) into J = −W + I02 K and taking the ﬁrst variation of J with respect to Ak , we arrive at the linear eigenvalue problem N )

(ωkl − λkkl )Al = 0 ,

k = 1, 2, . . . , N .

(7.5.61)

l=1

Explicit expressions for ωkl and kkl can easily be read of from (7.5.55) and (7.5.58). By a numerical solution of (7.5.61) the lowest eigenvalue λ is calculated. This λ depends on n and h and is related to the buckling current I0 through (7.5.60), yielding * + 2π 2 λ(h, n) πhR2 E I0cr = . (7.5.62) κ b2 (1 + ν)µ0 note: The coeﬃcient A2n does neither contribute to W nor to K; this mode represents a rigid-body translation of the spiral. Therefore, in case N ≥ 2n, we take A2n = 0. In the preceding three sections we have shown how one can ﬁnd the buckling current for helical or spiral superconductors. We shall present and discuss explicit numerical results in the next section. 7.5.4 Results We start with the result for the inﬁnite helix as given in (7.5.36). We would like to express this formula in standard form in terms of d and L. Here, the

7.5 Some Results for Superconducting Structures

253

choice for the distance 2d is clear, namely 2d = 2πh, but that for L is not so evident. There are two possibilities at our disposal: either L = πb, half the length of one turn, or L = 2πbn, the length between two supports. We prefer here the ﬁrst choice because we believe that this gives the best correspondence with the set of n parallel rods. With this, we can write (7.5.36) in the standard form (like in Table 7.3) * * dR2 N (n) E 2 , (7.5.63) I0cr = π (2n − 1)(1 + ν)κ L2 µ0 where κ and N (n) are given by (7.5.20) and (7.5.37), respectively. These deﬁnitions show that 2 4 R R R κ=1−2 1, (7.5.64) +O , for 2d 2d d while N (n) depends on ν, but only in a very weak sense; for ν running from 0 to 0.5, the value of N (n) changes less than 2%. Moreover, N (n) = 1 + O(n−2 ) ,

for

n1.

(7.5.65)

From this, we conclude that I0 is proportional to n−1/2 for large values of n. We recall that the direct Biot–Savard method yields a buckling current that can be found from (7.5.63) by taking κ = 1. For the ﬁnite helix, we have calculated the eigenvalues λ of the linear eigenvalue problem (7.5.48) numerically for n = 2, 3, . . . , 15. In so doing, we have used ν = 0 for the calculation of λ; this seems permissible because λ depends only very weakly on ν. The results of our calculations are presented in Table 7.4. In these calculations the following peculiarity was observed: the ﬁrst two coeﬃcients Ak of the eigenvector for any n were dominant, and it seems as if convergence was reached for N = 3 or 4 already, but for N = n a sudden jump in λ, small but apparent, occurs. Calculation of the eigenvectors showed that this jump occurred because the coeﬃcients An and An+1 were no longer negligible with respect to A1 or A2 . As a consequence, complete convergence was only found for N ≥ n + 2. Let us consider as an example the case n = 8. For this case we found for the normalized eigenvector A1 = −0, 94 ,

A2 = 0.25 ,

A3 = 0.02 ,

A8 = 0.21 ,

A9 = 0.15 , (7.5.66) while all remaining coeﬃcients are less than 10−2 . Analogous results were found for other values of n. From this, we conclude that the buckling mode for a ﬁnite helix consists of, ﬁrst, a global part, represented by A1 and A2 and, second, a local part on the scale of one winding, represented by An and An+1 . The latter part must be due to the direct interaction between two adjacent windings. Although this eﬀect is very striking with regard to the buckling mode, its eﬀect on the buckling value is no more than a few percent.

254

7 Magnetoelastic (In)stability and Vibrations

Table 7.4. Lowest eigenvalues λ × 102 (second, ﬁfth columns) for the set of numbers n (ﬁrst, fourth columns); the values in the third and sixth columns refer to λ = 1/(4nπ)(×102 ), see (7.5.67) n

λ × 102

1/4nπ × 102

n

λ × 102

1/4nπ × 102

2 3 4 5 6 7 8

3.87 2.71 2.05 1.62 1.34 1.13 .983

3.98 2.65 1.99 1.59 1.33 1.14 .995

9 10 11 12 13 14 15

.866 .773 .698 .636 .584 .539 .501

.884 .796 .723 .663 .612 .568 .531

Scrutiny of Table 7.4 leads us to the following observation: when we compare the calculated values of λ(n) in the second row of this table with those in the third row, we notice that the diﬀerence is always small; nowhere the relative diﬀerence exceeds 6%. Hence, for practical purposes it seems allowed to use for λ the very simple formula λ(n) =

1 . 4πn

(7.5.67)

Use of this formula in the expression for the buckling current (7.5.49) yields * + 2 dR π E I0cr = π 2 . (7.5.68) 2n(1 + ν)κ L2 µ0 Comparing the result (7.5.68) with the analogous result for an inﬁnite helix (7.5.63), we observe that in both cases the buckling current I0cr is proportional to n−1/2 for large values √ of n. However, the buckling current for a ﬁnite helix is always a factor π times that for an inﬁnite helix, no matter how large n may be. This result contradicts the expectation, logical at ﬁrst sight, which assumes that when n is large enough the formula for the inﬁnite helix also governs to a good approximation the buckling of a ﬁnite helix of n turns. This expectation was explicitly stated by Van de Ven and Lieshout in [256]. In the procedure for the inﬁnite helix, one complete period L − , ϕ ∈ [0, 4πn], was separated from the remaining part of the helix, L + . The expectation, stated in [256], was motivated by the assumption that the inﬂuence of the current in L + on the forces in L − becomes weaker for increasing n, and was practically restricted to a close neighbourhood of the end points of L − . If this would be true, the inﬂuence of L + on the buckling value would diminish and noninterference between L − and L + would be attained for n → ∞. However, the results of this section reveal that even for exceedingly large values of n the remote parts of the inﬁnite helix persist to

7.5 Some Results for Superconducting Structures

255

Table 7.5. The lowest eigenvalues λ × 102 for several values of n and h; the values in every second row refer to outcomes of formula (7.5.71)–(7.5.72) h

n=2

3

4

5

6

7

8

9

10

.050

11.78 10.97

6.00 5.97

3.82 3.88

2.70 2.77

2.04 2.11

1.61 1.67

1.31 1.37

1.10 1.15

.96 .98

.075

11.32 9.45

5.43 5.16

3.36 3.35

2.35 2.40

1.76 1.82

1.39 1.45

1.14 1.18

.95 .99

.83 .85

.100

10.91 8.54

5.00 4.65

3.05 3.02

2.12 2.16

1.59 1.64

1.26 1.30

1.03 1.07

.87 .89

.76 .76

.125

10.54 7.89

4.69 4.29

2.83 2.79

1.97 1.99

1.48 1.52

1.18 1.20

.97 .99

.82 .83

.72 .71

.150

10.24 7.38

4.45 4.02

2.67 2.61

1.86 1.87

1.40 1.42

1.12 1.13

.92 .92

.78 .77

.69 .66

interfere with the inner part L − . Hence, the replacement of the ﬁnite helix by an inﬁnite one, in order to get an easier problem, is never allowed. We have also computed the lowest eigenvalues λ = λ(h, n) of (7.5.61) for n running from 2 to 10 for several values of h; the results are presented in Table 7.5. In performing these computations, the results for the eigenvectors A showed us that the coeﬃcients Al for l close to 2n were dominant. Let us consider as an example the case h = 0.125 and n = 10. For this case we found for the normalized eigenvector A19 = −0, 657 ,

A21 = −0.513 ,

A18 = 0.435 ,

A17 = −0, 209 ,

A23 = −0.068 ,

A16 = 0.057 ,

A22 = 0.257, (7.5.69)

while all other coeﬃcients are less than 10−2 . Therefore, we changed the order of the summation in (7.5.38) in the following way: u(ϕ) =

N ) (2n − k)ϕ (2n + k)ϕ Aˆ2k−1 sin + Aˆ2k sin . (7.5.70) 2n 2n

k=1

This reordening improved the rate of convergence substantially; we did obtain a very satisfactory convergence already for N = 5. In [257, Fig. 6] one can ﬁnd plots of λ as a function of n and h. These graphs are on a doubly logarithmic scale and they give rise to an almost

256

7 Magnetoelastic (In)stability and Vibrations

linear behaviour in both h, for h ∈ [0.05 , 0.15] and for not too small values of n, i.e. n ≥ 4. This implies that λ(h, n) over this range must be of the form λ(h, n) = Λhα nβ .

(7.5.71)

By means of a least square approximation, we found Λ = 10.55 ,

α = −0.36 ,

β = −1.50 .

(7.5.72)

The results of this formula are also presented in Table 7.5 (as italic numbers). Everywhere in the range h ∈ [0.05 , 0.15] , n ≥ 4, the errors made by using this approximation are less than 5%. Hence, (7.5.71)–(7.5.72) represents a useful empirical formula for the buckling current. The latter then follows from * + 2 λ dR2 2π E , (7.5.73) I0cr = π 2 2 κ L µ0 where here we have taken L = πb1 = π(b0 + nπh). From the calculated values of the coeﬃcients Al of the eigenvector we can obtain an impression of the associated buckling mode. For explicit results and graphs of some of these buckling modes, we refer to [257, Fig. 7]; we only mention here that the displacements in the outer windings prevail over those in the inner ones. This could be expected by virtue of the greater mechanical stiﬀness of the inner windings as compared to the outer ones.

7.6 Magnetoelastic Vibrations of Superconducting Structures 7.6.1 Scope of this Section Up to here in this chapter, we have only considered static processes, related to magnetoelastic buckling. However, also dynamic processes, such as waves or vibrations, can be aﬀected by electromagnetic elastic interactions. For instance, magnetic ﬁelds or electric currents can inﬂuence the speed of propagation of waves or the eigenfrequencies or damping characteristics of vibrating magnetoelastic structures, and dynamic electromagnetic ﬁelds can generate forced elastic vibrations of such structures. On the other hand, elastic vibrations can change the electromagnetic ﬁelds in a magnetizable or electrically conducting body. The best known phenomenon in this is the piezoelectric eﬀect. Forced vibrations due to dynamic electromagnetic ﬁelds can have highly unwanted eﬀects as they generate noise (high acoustic sound pressures) or lead to resonance. Examples are in MRI-scanners, where acoustic noise is generated by the gradient coil; see e.g. Yao et al., [275], or in fusion reactors; see e.g. Tanaka,Tana1. For the analysis of forced vibrations, especially

7.6 Magnetoelastic Vibrations of Superconducting Structures

257

in connection with resonance, it is imperative to know the eigenfrequencies of the system. These eigenfrequencies are inﬂuenced by external, or bias, magnetic ﬁelds and electric currents, as we shall show in this section. As said before, electro-elastic vibrations are most well know from, and have found many practical applications in, piezoelectricity. An immens amount of literature is published on vibrations in piezoelectric systems, and it is far beyond the scope of this section to go deeper into this subject. Therefore, here we will not consider piezoelectric vibrations and we will restrict ourselves to mention only one review paper of Karlash, [107]. Magnetoelastic vibrations of electrically conducting plates and shells have been studied from the 1980’s on in the group around Ambartsumian, and later Bagdasarian, in the Yerevan University in Armenia. This research is continued nowadays by Hasanyan and his coworkers, [85, 120, 86], to mention only some recent papers from 2004–2005. These authors looked especially at vibrations of conducting strip-plates placed in an external magnetic ﬁeld. Hasanyan et al., [85], considered the free vibrations of soft ferromagnetic electrically conducting plates carrying an electric current. Nonlinear bending´rma ´n theory. For stretching vibrations are described by means of the von-Ka magnetoelastic interactions they used a model that looks like our Model IV. They found that the eigenfrequency ω depends on the bias magnetic ﬁeld H and current I such that ω increases with H (magnetic stiﬀening) but decreases with I, the latter up to a point I = Icr , where ω reduces to zero, implying that the basic state of the plate is unstable for I > Icr . Analogous results are found in the examples which we shall present in this section. Librescu et al., [120], considered free bending-stretching vibrations for perfectly conducting plates placed in an external magnetic ﬁeld H. Again, it is shown that ω increases with increasing H; this eﬀect becomes stronger for thinner plates. In 2005, Hasanyan et al., [85], reported on the eﬀect of ﬁnite versus inﬁnite conductivity. For the ﬁnite case, the eigenfrequency ω depends, besides on H, also on the conductivity σ. Moreover, special emphasis is given to the eﬀect of σ on the magnetic damping (this eﬀect is zero for perfectly conducting plates). The bending vibrations of a magnetoelastic beam on a spring foundation are analysed by Liu and Chang,Liu1. The beam is loaded by a mechanical axial force and a magnetic force. The analysis is based on the classical Moonapproach, [156]. Liu and Chang found that the magnetic ﬁeld reduces the deﬂection of the beam and decreases the eigenfrequency. Moreover, they registered the inﬂuence of the magnetic ﬁeld on the damping of the vibrating beam. A special paper is that by Kumar et al., [116], who studied the vibration control of a plate with a magnetostrictive layer by a coil surrounding the plate. The aim was to ﬁnd the location of the coil that gives the optimal damping; this optimal position depends on the mode shapes of the vibrating plate. The aim of this research was directed to smart structure design.

258

7 Magnetoelastic (In)stability and Vibrations

In this section, we present a selection of examples in which eigenfrequencies for speciﬁc electromagnetoelastic structures are determined. First, we calculate the eigenfrequencies of thin soft ferromagnetic circular plates placed in a transverse magnetic ﬁeld for diﬀerent support conditions of the plate; it is shown how the magnetic ﬁeld reduces the eigenfrequency until the buckled state is reached. The second example concerns a superconducting ring carrying a prescribed current; here, the current increases the eigenfrequency and, consequently, the ring is always stable. Finally, our variational principle is adapted for magnetoelastic vibrations of superconducting structures and applied to sets of two superconducting rings. By means of a kind of Rayleigh quotient the eigenfrequencies of these sets are calculated. It turns out that these eigenfrequencies decrease with increasing current up to a point where the system becomes unstable and buckles. We close this section by presenting a few more results for sets of parallel rods and for an inﬁnite helix. 7.6.2 Magnetoelastic Vibrations of a Thin Soft Ferromagnetic Circular Plate in a Uniform Transverse Magnetic Field In this section, we consider free ﬂexural vibrations of a thin circular plate placed in a uniform transverse magnetic ﬁeld B 0 . The plate is a soft ferromagnetic elastic body. For the description of the magnetoelastic vibrations of the plate, the Maxwell–Minkowski model will be used; see Section 3.4 (Model III). However, as far as the electromagnetic part is concerned we only consider quasi-static processes. For soft ferromagnetic media (with µ 1, say µ > 104 ), the magnetoelastic stresses according to the Maxwell– Minkowski model are approximately (for µ−1 ≈ 0) equal to the purely elastic (Hookean) stresses. In this model, also the magnetic body force is negligible (O(µ−1 )). The only signiﬁcant magnetoelastic coupling emanates from the boundary conditions. It will turn out that the magnetic ﬁelds generate a normal load per unit of area on the upper and lower surfaces of the deﬂected plate together with a shear force per unit of length at the edge of the plate. The plate has radius R and thickness 2h, where ε = h/R 1 for a thin plate. The z-axis of a system of cylindrical coordinates is chosen normal to the plate. The uniform basic ﬁeld is directed along the z-direction, so B 0 = B0 ez . For the rigid (undeformed) thin plate, we note that, except for a very small region (of O(ε)) near the edge r = R, the magnetic induction ﬁeld inside the plate is uniform and equal to B 0 . In using this approximation, we thus neglect the inﬂuences of the edge of the plate on the magnetic ﬁelds (we do not neglect the inﬂuence of the edge on the elastic stresses as we shall show further on); in other words, we assume for the magnetic problem the plate is inﬁnite in its plane. Our strategy to ﬁnd the eigenfrequencies of the plate now runs as follows: we split up the problem in a rigid-body part (whose solution we already know) and a perturbed part due to the dynamic deﬂections of the plates. The latter problem is then linearized with respect to

7.6 Magnetoelastic Vibrations of Superconducting Structures

259

the perturbations. This leads to a set of linearized balance and constitutive equations and boundary conditions; the general form of these relations for the Maxwell–Minkowski model can be found in Chap. 6 of this book. We shall neglect here all terms of O(ε); as one of the consequences, the magnetic body force vanishes and the Maxwell stress tensor becomes equal to the purely elastic tensor. These approximations result in the following set of equations for the magnetic potentials ψ , |z| > h, and ϕ , |z| < h, and the equation of motion for the stresses tij and the (vibrational) displacements ui = ui (x, t): ∆ψ = 0 ,

|z| > h ,

∆ϕ = 0 , tij,j = ρ¨ ui ,

|z| < h , |z| < h ,

ψ→0,

for |z| → ∞ , (7.6.1)

together with the boundary conditions on the upper and lower surface of the plate (here, B = B0 ; the stress boundary conditions at the edge r = R will be speciﬁed further on) ψ,r − ϕ,r = Bw (r) , txz = tyz = 0 ,

ψ,z − µϕ,z = 0 , µ tzz = − Bϕ,z , µ0

|z| = h ,

(7.6.2)

where w(r) is the rotationally symmetric normal deﬂection of the central plane of the plate. To be more precise, w(r) is here the amplitude part of the deﬂection; the total displacement as well as the perturbed potentials, are assumed to be time harmonic, i.e. proportional to eiωt , where ω is the real frequency. Hence, u(x, t) = w(r)eiωt ez (1 + O(ε)) .

(7.6.3)

We note that the right-hand side of the boundary condition for tzz in (7.6.2) originates from the rotation of the normal vector to the upper and lower surface of the plate; without this rotation these two terms would cancel each other. This boundary condition describes a plate loaded at its upper and lower surface by a normal stress. This is equivalent to the problem of a thin plate under a normal load per unit of area q(r), where q(r) = tzz |z=−h − tzz |z=h = −

&h & µ Bϕ,z && . µ0 z=−h

(7.6.4)

The classical Kirchhoff–Love theory then leads to the global equation of motion for the ﬂexural vibrations of a thin plate of the form D∆∆w(r) = q(r) − 2ρhω 2 w(r) , where D = 2Eh3 /3(1 − ν 2 ) is the plate constant.

(7.6.5)

260

7 Magnetoelastic (In)stability and Vibrations

We try to solve equations (7.6.1)1,2 for ψ (note that ψ = ψ(r, z) eiωt ) and ϕ by the separation of variables ψ(r, z) = Ψ (z)Ω(r) ,

ϕ(r, z) = Φ(z)Ω(r) .

(7.6.6)

The ﬁrst boundary condition then yields Ω (r) = w (r) ,

or

w(r) = Ω(r) + w0 ,

(7.6.7)

where w0 represents a rigid-body translation, which we will need further on to satisfy the mechanical (kinematical) boundary conditions at the edge of the plate; one should realize here that w0 is a dynamic rigid-body translation, since the actual displacement is w0 eiωt . It is evident that this rigid-body translation does not aﬀect the (quasi-static) magnetic potentials ϕ and ψ. The Laplace equations in (7.6.1)1,2 can now be split up into d2 1 d Ω(r) + λ2 Ω(r) = 0 , Ω(r) + dr2 r dr

and

d2 Φ(z) − λ2 Φ(z) = 0 , (7.6.8) dz 2

where λ is some real separation constant, which will follow from the mechanical boundary conditions at the edge r = R. There, it will turn out that λR = O(1), implying that λh = O(ε). Analogous equations can be found for Φ(z). The equation for Ω(r) is a Bessel equation, having the general solution Ω(r) = W J0 (λr) ,

(7.6.9)

while the solutions for Ψ (z), using that Ψ (z) → 0 for z → ∞, and for Φ(z), accounting for the symmetry in z, read Ψ (z) = C e−λ|z| ,

|z| > h ,

Φ(z) = D cosh(λz) .

(7.6.10)

The constants C and D and an expression for w(r) follow from the boundary conditions (7.6.2)1 , which yield, by using λh = O(ε) 1 and neglecting all O(ε)-terms except the term µλh (because µ is very large, µλh can be of O(1) for ε 1; in practice, we even have µε 1) C=

B µλh , 1 + µλh µ0

D=−

B 1 1 + µλh µ0

(7.6.11)

and w (r) = −λW J1 (λr). With these results we ﬁnd for the load q(r), deﬁned in (7.6.4), q(r) = −2µBΦ (h)Ω(r) =

2µλ2 h B 2 wJ ˆ 0 (λr) , µ0 (1 + µλh)

whereupon the equation of motion (7.6.5) becomes

(7.6.12)

7.6 Magnetoelastic Vibrations of Superconducting Structures

Dλ4 wJ ˆ 0 (λr) =

261

2µλ2 h B 2 wJ ˆ 0 (λr) + 2ρhω 2 (wJ ˆ 0 (λr) + w0 ) . (7.6.13) µ0 (1 + µλh)

If w0 = 0, this equation would immediately yield the dispersion relation for ω: Eh2 λ4 3µ(1 − ν 2 ) 2 2 ω = B . (7.6.14) 1− 3ρ(1 − ν 2 ) µ0 (1 + µλh)Eh2 λ2 However, in all of the explicit examples that we shall consider here, we have w0 = 0, and then the approach above does not yield a correct dispersion relation. However, we can already draw one general conclusion from this result, that remains valid even for w0 = 0. We see that ω 2 decreases with increasing B and becomes zero when the second term between the brackets on the right-hand side of (7.6.14) becomes equal to one. Let us denote by B0cr the associated value for B, and by ω0 the value for ω when B = 0. Then (7.6.14) can be written as B2 ω 2 = ω02 1 − 20 . (7.6.15) B0cr We thus see that if B > B0cr then ω 2 < 0 and ω becomes purely imaginary, implying that the unperturbed solution becomes unstable. Hence, B0cr is indeed the buckling value for B0 . Since the magnetic load q(r) in (7.6.12) and the elastic bending term in the right-hand side of (7.6.13) are not aﬀected by w0 , this result remained valid also for w0 = 0. However, w0 has its eﬀect on the inertia term in (7.6.13) and therefore the value of ω does depend on w0 . Since the analytical method derived above does not yield results for ω in the way we proposed it, we have to look for an alternative method. For this, we will adapt the variational method as introduced in Sect. 7.4.2, so as to include dynamical vibration problems, and after that we will apply Rayleigh’s Principle to ﬁnd the eigenfrequencies. To this end, we must calculate also the kinetic energy T and have to replace the Lagrangean L by the Hamiltonian H according to H = T − L. Because, as far as the electromagnetic part is concerned, we only consider quasi-static processes, the expressions found for the perturbed electromagnetic interaction integral K and the elastic energy W in the preceding sections remain the same here. Moreover, the kinetic energy is only due to the dynamic perturbed displacements and thus T is already the perturbed kinetic energy. Here, T is deﬁned on the perturbed displacement ﬁeld u = u(x, t) as 1 ˙ u)dV ˙ ρ(u, , (7.6.16) T = 2 G ˙ u) ˙ is the inner product of u. ˙ For steady-state vibrations with in which (u, (eigen)frequency ω, we can assume the displacements to be harmonic in time according to (7.6.3), whereupon T transforms into

262

7 Magnetoelastic (In)stability and Vibrations

R 1 2 ˆ )dV e2iωt = −ω 2 2πρh ω ρ(ˆ u, u rw2 (r)dr e2iωt 2 0 G = −ω 2 T. e2iωt . (7.6.17)

T =−

Applying Rayleigh’s Principle, we arrive at the well-known expression for the Rayleigh quotient, but now for magnetoelastic systems, W K 2 W −K B2 B0 = ω02 1 − 20 = 1− , (7.6.18) ω2 = W B0cr T. T. 2 and W/Tˆ = ω02 . We point out where we have used K = B02 K, W/K = B0cr here that this relation could also directly be derived from the global equation of motion (7.6.5) by multiplying this equation by w(r) and integrating the resulting equation over the plate. We will apply this method now to the soft ferromagnetic plate. We start with the expression for K according to (7.3.39), which with Bi = B(0) δi3 and u3 → Ω(r) becomes (we discard the e2iωt -term from now on) 1 ∂ B K=− 2 Bw(r) Ω(r) dS ψ− B ∂G ∂n µ0 R 2π =− rw(r)ψ,z (r, h)dr . (7.6.19) B 0

According to (7.6.6), the perturbed potential ψ(r, z) = Ψ (z)Ω(r) where Ψ (z) is given by (7.6.10)–(7.6.11) and w(r) = Ω(r)+w0 , with Ω(r) given by (7.6.9). Here it is important to note that (7.3.39) is derived under the assumption that µ−1 ≈ 0, inclusive (µλh)−1 ≈ 0, meaning that (7.6.11) then gives C ≈ B/µ0 . This leads us to 2π R 2πλ R K=− rw(r)Φ (h)Ω(r)dr = rw(r)Ω(r)dr . (7.6.20) B 0 µ0 0 For the elastic energy of a circular plate in rotationally symmetric bending we have; see [239, eq. (1.94)], W = πD

R

'

( r(∆w(r))2 − 2(1 − ν)w (r)w (r) dr .

(7.6.21)

0

For a clamped plate the contribution of the second term in the integrand for W vanishes, whereupon (7.6.21) reduces to

R

r(∆w(r))2 dr ,

W = πD

(7.6.22)

0

see [239, eq. (1.96)]. With (7.6.7)2 for w(r), expression (7.6.17) for the scaled kinetic energy Tˆ for the circular plate of thickness 2h becomes

7.6 Magnetoelastic Vibrations of Superconducting Structures

T. = 2πρh

R

r(w(r))2 dr = 2πρh 0

R

2

r (wJ ˆ 0 (λr) + w0 ) dr ,

263

(7.6.23)

0

an expression that clearly depends on w0 . We shall now illustrate this method by the following three examples. 1. Clamped plate In this case, the (kinematic) boundary conditions are w(R) = w (R) = 0 .

(7.6.24)

The second condition is met if J1 (λR) = J1 (Λ) = 0 ,

(7.6.25)

where Λ = λR, so Λ is dimensionless and of O(1). The lowest root of this equation is Λ1 = 3.832. Taking this mode as a good approximation for the buckling mode, i.e. taking Ω(r) = wJ ˆ 0 (Λ1 r/R), we obtain for the deﬂection w(r) r w(r) = w ˆ J0 Λ1 − J0 (Λ1 ) , (7.6.26) R where the ﬁrst condition of (7.6.24) is met by appropriately choosing the coeﬃcient w0 . We note that for the clamped plate w0 does not aﬀect the value of K or W , but it does aﬀect T.. Calculating K, W and T., we obtain successively πΛ1 R 2 2 2πλ 2 R 2 r dr = w ˆ rJ0 Λ1 w ˆ J0 (Λ1 ) , (7.6.27) K= µ0 R µ0 0 according to [1, eq.(11.3.34)] and with the use of (7.6.25) 4 R r Λ1 πD 4 2 2 2 W = πD dr = w ˆ rJ02 Λ1 Λ1 w ˆ J0 (Λ1 ) , (7.6.28) 2 R R 2R 0 and

r dr rJ02 Λ1 R 0 R r 1 +2J0 (Λ1 ) rJ0 Λ1 dr + R2 J02 (Λ1 ) R 2 02 2 2 2 = πρhR w ˆ J0 (Λ1 ) + 0 + J0 (Λ1 ) = 2πρhR2 w ˆ 2 J02 (Λ1 ) . (7.6.29)

T. = 2πρhw ˆ2

R

Substituting these results into (7.6.18), we arrive at 3 R πDΛ41 2 B02 2 2 2 ω = B 1 − = ω 1 − . 0 0 2 4πρhR4 µ0 D Λ1 B0cr

(7.6.30)

264

7 Magnetoelastic (In)stability and Vibrations

Here, Λ2 1 ω0 = √1 2 2 R

*

Λ2 h D = √1 2 2ρh 6 R

* E , ρ(1 − ν 2 )

(7.6.31)

is the eigenfrequency when B = 0. We ﬁnd good correspondence with [239, eq.(200)], who found as a ﬁrst approximation of the coeﬃcient in the √ ﬁrst right-hand side of (7.6.31) the value 10.33, whereas we ﬁnd Λ21 / 2 = 10.37 (here one should realize that Timoshenko considers plates of thickness h instead of 2h as we do). To get a better correspondence with the more precise value 10.21 (see [239, eq. 201]), we have to take for w(r), or Ω(r), a Fourier–Bessel series according to w(r) = Ω(r) =

N ) 1

r , wk J0 Λk R

(7.6.32)

with Λk the roots of J1 (Λk ) = 0, and were the last coeﬃcient wN is chosen such that w(R) = 0. The best coeﬃcients wk , k = 1, 2, . . . , N − 1 can then be found by equating ∂ω 2 /∂wk = 0 for k = 1, 2, . . . , N . We can use the result (7.6.30) also to obtain directly the value for the magnetic ﬁeld for which the ferromagnetic plate buckles. The value for the buckling ﬁeld obtained in this way is 3/2 3/2 1 Λ1 h h B0cr 4.33 √ = = . (7.6.33) 2 2 R µ0 E 3(1 − ν ) (1 − ν ) R This result is in agreement with the buckling ﬁeld found by Van de Ven, [250, eq. (4.11)]. 2. Simply supported plate Here, besides the kinematical boundary condition w(R) = 0 also a dynamical condition, expressing that the bending moment along the edge should be zero, holds. However, the latter only holds provided no bending moment of magnetic origin is acting at this edge. This is diﬃcult to prove here, because for this we need the mechanical boundary conditions for the stresses at the edge, and for these we need the magnetic ﬁelds there. And this is exactly what we do not know, because at the edge either the magnetic ﬁelds become singular or the magnetization becomes saturated. Therefore, we looked for another way. In [250], we showed by using expressions for the total force and the total moment on the plate and requiring that they should be in equilibrium with the total bending moment and shear force along the edge, that indeed the bending moment of magnetic origin is zero along the edge (this in contrast to the shear force of magnetic origin as we shall see in the next case). Thus we obtain the classical boundary condition ν w (R) = 0 , (7.6.34) w (R) + R

7.6 Magnetoelastic Vibrations of Superconducting Structures

265

yielding

(1 − ν) J1 (λR) = 0 , λR of which the lowest root for ν = 0.3 is J0 (λR) −

Λ1 = λ1 R = 2.049 .

(7.6.35)

(7.6.36)

By adding the rigid-body translation w0 , we obtain exactly the same relation for w(r) as in (7.6.26). In an analogous way as we did for the clamped plate we can now calculate the integrals K, W and Tˆ from the same expressions as found in (7.6.27)–(7.6.29), but the outcomes diﬀer somewhat due to the diﬀerent boundary condition (7.6.34) (now J1 (Λ1 ) = (Λ1 /(1 − ν))J0 (λ) = 0). Thus we obtain (use [1, eq.(11.3.34)]) 2πΛ1 R rw(r) wJ ˆ 0 (λ1 r) dr K= µ0 R 0 ! R R r r 2πλ 2 2 = dr − J0 (Λ1 ) dr w ˆ rJ0 Λ1 rJ0 Λ1 µ0 R R 0 0 πΛ1 R 2 2 2 2 = w ˆ J0 (Λ1 ) + J1 (Λ1 ) − J0 (Λ1 )J1 (Λ1 ) µ0 Λ1 πΛ1 R = w ˆ 2 Λ21 − (1 − ν 2 ) J02 (Λ1 ) , (7.6.37) 2 µ0 (1 − ν) where in the last step we have used (7.6.35) to eliminate J1 (Λ1 ) in favour of J0 (Λ1 ), ! Λ41 2 R 2 r 2 dr − (1 − ν)(w (R)) w ˆ rJ0 Λ1 W = πD R4 R 0 πD 2 4 2 w ˆ Λ1 (J0 (Λ1 ) + J12 (Λ1 )) − 2(1 − ν)Λ21 J12 (Λ1 ) 2 2R πDΛ41 w ˆ 2 Λ21 − (1 − ν 2 ) J02 (Λ1 ) , (7.6.38) = 2 2 2(1 − ν) R

=

and, ﬁnally, R r r rJ02 Λ1 rJ0 Λ1 dr − 2J0 (Λ1 ) dr R R 0 0 1 2 2 + R J0 (Λ1 ) 2 ' 2 ( 2 1 2 2 2 1 2 = 2πρhR w J (Λ1 ) + J1 (Λ1 ) − ˆ J0 (Λ1 ) J1 (Λ1 ) + J0 (Λ1 ) 2 0 Λ1 2 2πρhR2 = w ˆ 2 (Λ21 − 2(1 − ν 2 )) J02 (Λ1 ) . (7.6.39) 2(1 − ν)2

T. = 2πρhw ˆ2

R

266

7 Magnetoelastic (In)stability and Vibrations

Again substituting these results into (7.6.18), we arrive at 3 2 R πDΛ41 Λ21 − (1 − ν 2 ) 2 1− B02 ω = 2πρhR4 Λ21 − 2(1 − ν 2 ) µ0 D Λ1 B2 = ω02 1 − 20 , (7.6.40) B0cr where, *

Λ21 − (1 − ν 2 ) Λ21 − 2(1 − ν 2 ) * E h , = 2.987 2 R ρ

Λ2 ω0 = 12 R

*

* D 4.936 = 2ρh R2

D 2ρh (7.6.41)

is the eigenfrequency when B = 0. For the buckling value, we ﬁnd the same expression as in (7.6.33), but due to the diﬀerent value for Λ1 we obtain here, for ν = 0.3, B √ 0cr = 1.78 µ0 E

h R

3/2 .

(7.6.42)

This result is again in agreement with the buckling ﬁeld found by Van de Ven, [250, eq. (4.12)]. We still want to note here that in the calculation of K in (7.6.37), the ˆ 0 (Λ1 ) aﬀects the outcome for K (i.e. rigid-body translation w0 = wJ replacing w(r) by Ω(r) in the ﬁrst integral would result in a diﬀerent (wrong) value for K). This may look at ﬁrst sight somewhat surprising, but this eﬀect is due to the presence of a shear force of magnetic origin at the edge of the plate, as we shall see in the next example. Due to this shear force, the rigid-body translation performs work and thus contributes to the magnetic interaction integral K. This is not actually so here, because in this case the displacement of the edge is zero, and thus the work done by the shear force at the edge is now zero (but this is only so because we included the rigid-body translation w0 in our choice for the displacement w(r)). However, this is no longer true in the next example; we will show there how to incorporate the work done by the shear force in the magnetic interaction integral K. 3. Free plate For the free plate, one would expect as dynamic boundary conditions for this rotationally symmetric problem that the bending moment and the shear force are zero at the edge. However, only the ﬁrst condition is correct here. For the second one, we calculate the total force due to the magnetic ﬁeld on the plate, which because of equilibrium should be equal to zero. This force is composed of the force due to the normal load q on

7.6 Magnetoelastic Vibrations of Superconducting Structures

267

the surface of the plate plus the force due to possible shear forces at the edge. Let us denote this shear force by Q, then we have for the total force in the z-direction R q(r)rdr = 0 . (7.6.43) Fz = 2πRQ + 2π 0

This leaves us with a shear force equal to (for µλh ≈ 0) 2λ B2w Q=− ˆ µ0 R

R

rJ0 (λr)dr = − 0

2Λ B 2 wJ ˆ 1 (Λ) . µ0 R2

The boundary conditions at the free edge then become d Q ν w (R) + w (R) = 0 , ∆w (R) = − . R dr D

(7.6.44)

(7.6.45)

Moreover, we now choose the rigid-body translation w0 such that the total momentum of the plate becomes zero, i.e.

R 0

R

rJ0 (λr)dr −

rw(r)dr = w ˆ 0

1 w0 R2 2

R2 1 J1 (Λ) − w0 R2 = 0 , =w ˆ Λ 2

(7.6.46)

or

2 J0 (Λ) w ˆ. (7.6.47) Λ With this choice, the second boundary condition of (7.6.45) follows immediately from one integration of the equation of motion (7.6.13) over the surface of the plate. We then only have to satisfy the ﬁrst boundary condition, which, as in the preceding case, is satisﬁed by choosing Λ = Λ1 = 2.049, for ν = 0.3. The choice (7.6.47) for w0 yields for the deﬂection of the plate 2 r J1 (Λ1 ) , (7.6.48) w(r) = w ˆ J0 (Λ1 ) − R Λ1 w0 =

and thus w(R) = 0 now. This implies that the work done by the shear force Q at the edge becomes unequal to zero and thus must be incorporated into K. The easiest way to incorporate this work into K is by replacing w(r) by w(r) − w(R) in the formula for K, (7.6.19), yielding (compare also with (7.6.27)) 2πλ 2 w ˆ K= µ0

0

R

r r J0 Λ1 − J0 (Λ1 ) dr , rJ0 Λ1 R R

(7.6.49)

268

7 Magnetoelastic (In)stability and Vibrations

the same expression as in the second example. Since also the elastic energy remains the same, this implies that we will here ﬁnd the same buckling value as in the preceding example, and as in [250, eq. (4.12)]. However, as the kinetic energy depends on the rigid-body translation w0 , the frequency will diﬀer from the value found in the preceding example. Here we obtain

R r r 4J1 (Λ1 ) dr − dr rJ02 Λ1 rJ0 Λ1 R Λ1 R 0 0 2R2 J12 (Λ1 ) + Λ21 ' 2 ( 2 2 2 2 1 2 = 2πρhR w J (Λ1 ) + J1 (Λ1 ) − 2 J1 (Λ1 ) ˆ 2 0 Λ1 ' ( 2πρhR2 = w ˆ 2 Λ21 + (1 − ν)2 − 4 J02 (Λ1 ) . (7.6.50) 2(1 − ν)2

T. = 2πρhw ˆ2

R

Again substituting these results into (7.6.18), we arrive at 3 4 2 2 Λ − (1 − ν ) 2 R πDΛ 1 1 1− ω2 = B02 2πρhR4 Λ21 + (1 − ν)2 − 4 µ0 D Λ1 B02 2 = ω0 1 − 2 , (7.6.51) B0cr where, *

* Λ21 − (1 − ν 2 ) D Λ21 + (1 − ν)2 − 4 2ρh * * h D E 9.178 = 5.299 2 , = 2 R 2ρh R ρ

Λ2 ω0 = 12 R

(7.6.52)

is the eigenfrequency when B = 0. All three examples considered above show the same behaviour: the eigenfrequency decreases with increasing magnetic bias ﬁeld B0 up to a value B0cr when ω becomes zero, indicating that the undeﬂected state of the plate becomes unstable. Moreover, we have seen that a shear force of magnetic origin is acting at the edge of the plate, which can have a dominant eﬀect on the frequency relation. The latter eﬀect is strictly inherent to the Maxwell– Minkowski model used here as we shall see in the following, where we compare the Maxwell–Minkowski model with the Lorentz model (Model V in Chap. 3 of this book).

7.6 Magnetoelastic Vibrations of Superconducting Structures

269

Comparison between the Maxwell-Minkowski and Lorentz models. The equations derived above are based on the Maxwell–Minkowski model (Model III). We now shall prove that two theories based on diﬀerent models, here Model III and the Lorentz model (Model V) (this model is in the liter`rean current model by a.o. Moon) yield identical ature also called the Ampe results. This is of course completely in agreement with the spirit behind this book. From Chap. 4, Sects. 4.4–4.5, more speciﬁcally, from equations (4.4.2) and (4.5.4) we can deduce the following relation between the stresses in Models III and V: V tij = III tij + Mi Bj + 12 δij Mk (Bk + Hk ) , (7.6.53) which for a soft ferromagnetic medium and with the same approximations as used throughout this section, reduces to V

tij = tij −

1 1 Bi Bj + δij Bk Bk , µ0 2µ0

(7.6.54)

where we have used that III tij ≈ tij , the purely elastic stress. Hence, we see here already one important diﬀerence in the two formulations: whereas in the constitutive equation for III tij the magnetic part is negligible, for V tij this part is of the same order of magnitude as the elastic part. After linearization of (7.6.54), we obtain for the perturbed stresses V

tij = tij − µB(δi3 ϕ,j + δj3 ϕ,i − δij ϕ,z ) .

(7.6.55)

Analogously to (7.6.1) and (7.6.2) this leads to the equation of motion and the boundary conditions in terms of the Lorentz stresses V

tij,j = ρ¨ ui ,

|z| < h ,

(7.6.56)

and V

|z| = h .

ti3 = µB(−δi3 ϕ,z + ϕ,i ) ,

(7.6.57)

By comparing this with (7.6.1) and (7.6.2), we see that the equation of motion is the same, but that the boundary conditions are diﬀerent. This will lead to a diﬀerent relation for the shear force as we shall show now. However, ﬁrst we show that the relation for the bending moment does not change. The bending moment is deﬁned as V

Mrr =

h

z −h

V

tzz dz = Mrr −

µB µ0

h

−h

zϕ,z dz ,

(7.6.58)

where the last step follows after substitution of (7.6.55) and where Mrr is the purely elastic bending moment given by its constitutive equation ν Mrr = −D w (r) + w (r) . (7.6.59) r

270

7 Magnetoelastic (In)stability and Vibrations

For the shear force V

V

Q we ﬁnd analogously

h

Q=

z

µB d tzr dz = Q + µ0 dr

V

−h

h

ϕ(r, z)dz ,

(7.6.60)

−h

with

d ∆w(r) . (7.6.61) dr If we want to compare the orders of magnitude of the magnetic terms in V Mrr and V Q, we must compare V Mrr,r with V Q, or Q = −D

h

−h

zϕ,z dz

h

with

ϕdz . −h

Using (7.6.6) and (7.6.9), we immediately see that

h

−h

zϕ,z dz = O((λh) )

h

2

h

2

ϕdz = O(ε ) −h

ϕdz ,

(7.6.62)

−h

from which we conclude that the magnetic part in or V Mrr = Mrr .

V

Mrr may be neglected, (7.6.63)

On the other hand, for the shear force this implies that the magnetic term in (7.6.60) may not be neglected as it is of the same order as the elastic term. In the common way, by integrating the local equations of motion (7.6.56) over the thickness of the plate and using in this process the constitutive equation (7.6.55) and the boundary conditions (7.6.57), we can derive the global equation of motion (compare with (7.6.5)) D∆∆w(r) = q(r) − 2ρhω 2 w(r) , where now q(r) =

µB µ0

d2 1 d + dr2 r dr

(7.6.64)

h

ϕ(r, z)dz ,

(7.6.65)

−h

but, as follows from ∆ϕ(r, z) = 0, this can also be written as &h & µB ϕ,z && , q(r) = − µ0 z=−h

(7.6.66)

which is identical to (7.6.4). Finally, we look at the boundary conditions for a free plate. In the same way as we derived (7.6.44) from (7.6.43) we obtain here for the shear force at the edge V Q=0. (7.6.67)

7.6 Magnetoelastic Vibrations of Superconducting Structures

271

Using (7.6.6) and (7.6.9)–(7.6.11) together with (µλh)−1 1 and λh 1, we derive from (7.6.60) h ˆ d ˆ B2w 2B 2 w V Q= cosh(λz)dz J0 (λr) = J1 (λr) . (7.6.68) µ0 λh dr µ 0 −h where Q is the purely elastic shear force. This leads us to the boundary conditions h d µB d ν w (R) = 0 , ∆w (R) + w (R) + −D ϕ(r, z)dz = 0 , R dr µ0 dr −h (7.6.69) which are exactly the same conditions as (7.6.45). Thus, we have proved that the Lorentz formulation for this problem is completely equivalent to the Maxwell–Minkowski formulation. This proof can be extended to the other models presented in Chap. 3 of this book. 7.6.3 Magnetoelastic Vibrations of a Superconducting Ring in its Own Field In this section, we investigate the in-plane vibrations of a circular ring carrying an electric current, which is conﬁned to the surface of the superconducting ring; see [254]. The investigation is based on a perturbation method: the ﬁelds in the ﬁnal, deformed, state of the ring are decomposed into the ﬁelds for the undeformed ring, the rigid-body ﬁelds, and the perturbations on these ﬁelds. The latter are due to the dynamic deﬂections of the ring. However, as far as the electromagnetic part is concerned we only consider quasi-static processes. First, the rigid-body problem is solved. As a speciﬁc result the initial stresses due to the magnetic forces (of Lorentz type) in the undeformed coil are obtained; it will turn out that these stresses play an essential role in the calculation of the eigenfrequencies of the ring. Next, the linearized perturbed problem is solved, yielding expressions for the load of magnetic origin on the deformed ring. By superposition, the total load on the deformed ring is now known, and from this an equation of motion for the vibrational in-plane motion of the slender ring is derived. The solution of this equation leads to an expression for the eigenfrequencies of the superconducting ring. This expression consists of two terms: one due to the initial stresses and one due to the perturbed ones. The ﬁrst term increases the frequency with increasing current (and, hence, has a stabilizing eﬀect), whereas the second term causes the frequency to decrease (a destabilizing eﬀect). We will show that the ﬁrst term dominates the second one, thus implying that the undeformed state of the ring is stable against in-plane vibrations. This result agrees with that of Chattopadhyay [41], who obtained this result by purely numerical means. The ring that is considered here is an elastic superconducting slender ring of radius b and circular cross-section, radius R; the ring is called slender if

272

7 Magnetoelastic (In)stability and Vibrations

ε = R/b 1. The ring is placed in a vacuum and carries an electric surface current J at its surface, the total current being I0 . The magnetic ﬁeld inside the ring is zero, and that outside the ring has to satisfy eijk Bk,j = 0 ,

Bi,i = 0 ,

B → 0 , |x| → ∞ .

(7.6.70)

The current density J is related to the boundary value of the external ﬁeld B through the boundary conditions at the boundary ∂G of the ring µ0 Ji = −eijk Bj nk ,

Bi ni = Ji ni = 0 ,

x ∈ ∂G .

`re’s law The total current I0 is prescribed by means of Ampe (B · ds) = µ0 I0 ,

(7.6.71)

(7.6.72)

C

where C is a contour entirely encircling a cross-section of the conducting ring. The Cauchy stresses Tij have to satisfy the equations of motion ¨i , Tij,j = ρU

(7.6.73)

where U is the total displacement vector. In the Maxwell–Minkowski model the current J produces a surface tension at ∂G according to 1 1 1 (7.6.74) Bi Bj − δij Bk Bk = eijk Jj Bk , x ∈ ∂G . Tij nj = µ0 2 2 The above equations hold on the unknown, deformed state of the ring and are thus nonlinear. To derive a linear system, we ﬁrst suppose the existence of an intermediate static equilibrium state. Since for our considerations the deformations in this state are not so relevant (this in contrast with the prestresses in this state), we approximate this state by the rigid-body state. The ﬁelds Bi0 and Tij0 in this state satisfy the same equations (7.6.70)–(7.6.74) as above, but now in the known reference conﬁguration. For later use, we especially need the equation for the traction at ∂G Tij0 nj = Ti0 = 12 eijk Jj0 Bk0 .

(7.6.75)

The perturbation on the rigid-body state is characterized by the displacement vector u = u(x, t). The perturbed magnetic ﬁelds may be considered as being time-harmonic functions of x and t. These ﬁelds are decomposed as B = B 0 (x) + b(x)eiωt , J = J 0 (x) + j(x)eiωt and Tij = Tij0 (x) + tij (x)eiωt (the harmonic term eiωt will be discarded in the quasi-static electromagnetic equations). The linearization procedure now results in the following sets of equations (from here on we omit the the upper 0 )

7.6 Magnetoelastic Vibrations of Superconducting Structures

273

inside the ring ui ; tij,j − Tij,k uk,j = ρ¨

(7.6.76)

in vacuum eijk bk,j = 0 ,

bi,i = 0 ,

b → 0 , |x| → ∞ ;

(7.6.77)

at the surface ∂G of the ring eijk bj nk + eijk Bj,l ul nk − eijk Bj ul,k nl = −µ0 ji + µ0 Ji uj,k nj nk , bi ni = Bi uj,i nj − Bi,j uj ni , ji ni = Ji uj,i nj , tij nj = Ti uj,k nj nk + Tij uk,j nk + ti ,

(7.6.78)

where ti = 12 eijk jj Bk + 12 eijk Jj (bk + Bk,l ul ) ,

(7.6.79)

and the incremental stresses tij are given by tij = −Tij uk,k + Tjk ui,k + Tik uj,k + τij ,

(7.6.80)

where τij is the incremental elastic stress, which is directly related to the inﬁnitesimal deformations by Hooke’s law. Here, one must realize that all these equations refer to the undeformed reference state of the ring. ˆ (r, φ, z)eiωt be the time-harmonic displacement of Let u = u(x, t) = u the ring, where {r, φ, z} are cylindrical coordinates in the centre of the ring. Since we consider in-plane vibrations only, the displacement of the central line of the slender ring can be presented by (compare with (7.4.47)) ' ( (r − b) ˆ (r, φ, z) = w(φ)er + v(φ) − u (w (φ) − v(φ)) eφ 1 + O(ε2 ) . b (7.6.81) For an inextensible ring, the displacements are restricted by the constraint v (φ) + w(φ) = 0 .

(7.6.82)

In [254], the rigid-body problem and the linear perturbed problem for the magnetic ﬁeld and the current are solved by introducing a separation of variables in toroidal coordinates. We do not repeat here these mathematical manipulations, which result in expressions for the mechanical body forces and the tractions at ∂G in the rigid-body state and the perturbed state. Integration of the local equations of motion (7.6.76) over the cross-section of the ring, with use of the boundary conditions (7.6.78)4 , leads us to the global equation of motion for the in-plane vibrations of the ring. These global equations can also be found in [156, eqs. (6-7.3)-(6-7.4)]; for in-plane vibrations uy = Ny = Gx = H = 0. Moreover, in our notations, ux = −w , uz = v.

274

7 Magnetoelastic (In)stability and Vibrations

After the elimination of Nx and T and with the use of the well-known constitutive equation for the bending moment Gy in an inextensible slender ring, Gy = −

EI πER4 (w (φ) + w(φ)) = − (w (φ) + w(φ)) , b2 4b2

(7.6.83)

one obtains πER4 v (w (φ) + 2w (φ) + w (φ)) = Γ + πρω 2 b2 R2 (w (φ) + v(φ)) . (7.6.84) 4b2 Here, Γ is a load parameter, arising from the body force in (7.6.76) and the traction (7.6.78)4 , and given by Γ (φ) = bKr (φ) + bKφ (φ) + L (φ) + L(φ) ,

(7.6.85)

where Kr and Kφ are forces per unit of length in r- and φ-direction, respectively, and L is a (bending) moment per unit of length about the z-axis. As is shown in [254], the contribution of the moment L to Γ is of order ε2 compared to the other terms, and thus negligible (L(φ) = L (φ) = 0), while the remaining part can be split up according to Γ = Γ1 + Γ2 , where Γ1 is the rigid-body contribution and Γ2 the perturbation part, with 8 3µ0 I02 1 w (φ) log Γ1 (φ) = − − (1 + O(ε log ε)) , (7.6.86) 4π ε 2 and 8 3µ0 I02 17 w (φ) log − (1 + O(ε log ε)) . Γ2 (φ) = 4π ε 6

(7.6.87)

From these results we conclude that the highest-order contributions of Γ1 and Γ2 (of O(log ε)) are equal but opposite, so they cancel each other. This implies that the remaining part of Γ = Γ1 + Γ2 is of O(ε0 ) = O(1), and equal to 7µ0 2 I w (φ)(1 + O(ε log ε)) . Γ (φ) = − (7.6.88) 4π 0 As in most preceding examples in this chapter, the separation of variables in toroidal coordinates requires the relation w (φ) + λ2 w(φ) = 0 ,

(7.6.89)

to be satisﬁed. Taking account of the periodicity in φ and excluding rigidbody translations, we see that the lowest eigenmode is represented by λ = 2, yielding w(φ) = W cos 2φ . (7.6.90) Then, according to the inextensibility condition (7.6.82),

7.6 Magnetoelastic Vibrations of Superconducting Structures

v(φ) = − 12 W sin 2φ .

275

(7.6.91)

Substituting (7.6.90) and (7.6.91) into the global equation of motion (7.6.84), we arrive at the dispersion relation for ω 18πER4 7µ0 2 5 I + πρb2 R2 ω 2 . =− 2 4b 2π 0 2

(7.6.92)

*

With the deﬁnition R ω02 = 2 b

9E , 5ρ

(7.6.93)

the eigenfrequency of the free ring (I0 = 0), the dispersion relation (7.6.92) yields for the eigenfrequency 7µ0 b2 2 ω 2 = ω02 1 + 2 I . (7.6.94) 9π ER4 0 Hence, the frequency of the current-carrying ring increases with increasing current. For a conservative problem, as that considered here, instability occurs when ω becomes zero. Since the right-hand side of (7.6.94) remains positive for all values of I0 , we infer that the equilibrium state of the ring (i.e. the pre-stressed rigid-body state) is always stable against in-plane vibrations. In analyzing the eﬀects of Γ1 and Γ2 separately, we observe that Γ1 , due to the initial unperturbed ﬁeld, has a stabilizing eﬀect, whereas Γ2 , due to the perturbed ﬁeld, has a destabilizing eﬀect. When taken together, the stabilizing eﬀect dominates over the destabilizing one, and therefore the ring remains stable. The out-of-plane vibrations can be treated in an analogous way, and there is no reason to expect that this will yield essentially diﬀerent results. This is conﬁrmed by the results of [41]. In order to illustrate the inﬂuence of the current I0 on the eigenfrequency ω quantitatively, we use the following numerical values (these values are from [41]) 2

E = 8 × 1010 N/m , R = 3.03 m ,

I = 2.2 × 10−4 m4 , µ0 = 4π × 10−7 H/m .

For these values, (7.6.94) becomes ' ( ω 2 = ω02 1 + 4.06 × 10−14 I02 .

(7.6.95)

(7.6.96)

This result agrees, at least in order of magnitude, reasonably well with that of Chattopadhyay [41, eq. (42)]. In this aspect it must be mentioned that in [41] it was assumed that the current is uniformly distributed over the cross-section. This correspondence between our results and those of [41] suggests that the speciﬁc distribution of the current over the cross-section (i.e.

276

7 Magnetoelastic (In)stability and Vibrations

a uniform distribution in [41] against a distributed surface current in our approach) can at most be of quantitative inﬂuence and not qualitative. A diﬀerent current distribution will change the numerical value of the coeﬃcient of I0 in the dispersion relation for ω, but it is not to be expected that it can disturb the stability of the ring. 7.6.4 Variational Principle for Magnetoelastic Vibrations of Superconducting Structures In this section, we will demonstrate how we can generalize our variational method presented in the preceding sections to include also vibrational problems for superconducting structures, and how we can ﬁnd the eigenfrequencies of these structures from this. The basic idea for this is rather simple: we only replace the Lagrangean L introduced in Sect. 7.4 by the Hamiltonian H = T − L, where T is the kinetic energy. However, in doing this we shall as far as the electromagnetic part is concerned only consider quasistatic processes. This also holds for the perturbed part and, therefore, the expressions found for the perturbed electromagnetic interaction integral K in the preceding sections remain the same here. Of course, this also holds for the elastic energy. Moreover, the kinetic energy is only due to the perturbed displacement and thus T is already the perturbed kinetic energy. Therefore, we have to formulate here only the Hamiltonian for the perturbed case (which is in fact the second variation of the full Hamiltonian) and we ﬁnd for this H = T − (W − K) = T − W + K. Here, T is deﬁned on the perturbed displacement ﬁeld u = u(x, t) as 1 ˙ u)dV ˙ ρ(u, . (7.6.97) T = 2 G For steady state vibrations with (eigen)frequency ω we can assume the displacements harmonic in time according to ˆ (x) eiωt , u(x, t) = u

(7.6.98)

whereupon T transforms into ω2 T =− 2

ˆ )dV = −ω 2 Tˆ . ρ(ˆ u, u

(7.6.99)

G

The equation of motion and the electromagnetic equations for the perturbed ﬁelds follow from equating the ﬁrst variation of H to zero, but since H is a homogeneous quadratic functional in the perturbations, then also H = 0 in the perturbed state (analogous to J = 0 in the magnetostatic case). Thus, putting H equal to zero, we arrive at the well-known Rayleigh quotient, but now for magnetoelastic superconducting systems, in the form

7.6 Magnetoelastic Vibrations of Superconducting Structures

W −K = ω02 ω = ˆ T 2

1−

I0 I0cr

277

2 ,

(7.6.100)

where in the latter step we have used K = I02 K and K/W = (I0cr )2 , and where we have introduced ω02 = W/Tˆ. Since we have derived values for I0cr for various systems in the preceding sections, we can, from the relation above, directly read oﬀ the dependencies of the eigenfrequencies on the imposed current I0 . We conclude that for all such systems the eigenfrequency decreases from ω0 (in case I0 = 0) to a value going to zero for I0 → I0cr . For I0 > I0cr , ω 2 becomes negative, meaning that the eigenfrequency becomes purely imaginary, implying instability of the system. Hence, this frequency analysis conﬁrms that the system buckles whenever I0 > I0cr . Example: A set of two concentric superconducting rings We consider here the same system as described in Sect. 7.4.2; see also Fig. 7.5. To start with the kinetic energy, we describe the displacement as in (7.6.98) ˆ (i) , the deﬂection of the central line of the i-th ring, given by (7.4.47). with u This yields the following expression for T (i) , the kinetic energy of the i-th ring, 2 ! ω2 r − bi (i) 2 T =− wi (φ) + vi (φ) − ρdV (wi (φ) − vi (φ)) 2 G(i) bi 2π ' 2 ( ' ( πω 2 2 wi (φ) + vi2 (φ) dφ 1 + O(ε2 ) , ρbi R =− (7.6.101) 2 0 where in the latter step we have used that the integral of (r − bi ) over G(i) is zero, while the integral with (r − bi )2 yields a term of O(ε2 ) where ε = R/b and b = (b1 + b2 )/2. Use of, see (7.4.50), wi = ±w cos(2φ) and vi = ∓ 12 w sin(2φ), such that the inextensibility condition vi (φ) + wi (φ) = 0 is fulﬁlled, leads us to T (i) = − 58 πω 2 ρbi R2 w2 .

(7.6.102)

This ﬁnally gives for the total kinetic energy T , or better Tˆ, Tˆ = 54 πρbR2 w2 (1 + O(ε)) ,

(7.6.103)

where we have used that bi = b (1 + O(ε)) for i = 1, 2. With the use of relations (7.4.51) and (7.4.52) for W and K, respectively, in (7.6.100) we then arrive at the relation for the eigenfrequency of a set of two concentric superconducting rings 2 ! 9ER2 W − I02 K µ0 Q b2 I0 2 = ω = 1− . (7.6.104) 5ρb4 E 3πR3 Tˆ

278

7 Magnetoelastic (In)stability and Vibrations

A few more results We can apply the method above also to some of the other examples dealt with in the preceding section. We only present the results for the eigenfrequencies here. 1. A pair of two parallel rods E ω2 = ρ

π2 R 2L2

2

µ0 Q 1− E

L2 I0 π 3 R3

2 ! .

(7.6.105)

2. A pair of two parallel rings 9E ω = 5ρ

2

R b2

2

µ0 (5 + ν)Q 1− E

b2 I0 6πR3

2 ! .

(7.6.106)

3. A set of n parallel rods E ω = ρ 2

π2 R 2L2

2

µ0 Q 1− E

L2 I0 αn R3

2 ! .

(7.6.107)

4. An inﬁnite helix periodically supported at every n-th turn EN (n) ω = 8(1 + ν)ρ 2

R nb2

2

µ0 κ(1 + ν)(2n − 1) 1− EN (n)

b2 I0 πhR2

2 ! . (7.6.108)

8 Electrorheological Fluids

8.1 Introduction “Smart” materials can adaptively change or respond to an external stimulus producing a useful eﬀect. Mechanical stresses, temperature, electric or magnetic ﬁelds, photon radiation or chemicals are typical examples of stimuli. A useful eﬀect usually means a dramatic change of either one physical property (mechanical, electrical, appearance), the structure or the composition, which can be monitored and used in certain applications. A useful eﬀect may be completely reversed when the stimulus is removed and this important feature permits an easy control through simply changing the environmental conditions. A variety of smart materials exists, which are being researched extensively. These include piezoelectric and thermoelectric materials, magnetorheological and electrorheological ﬂuids, photochromic and thermochromic materials, electroluminiscent, ﬂuorescent and phosphorescent materials and shape memory alloys. Electrorheological ﬂuids (often abbreviated as ERF) are such intelligent materials that exhibit drastic changes in their rheological properties upon the application of an outer electric ﬁeld on the order of 1 kV/mm. The ER phenomenon is characterized by full reversibility and a very fast response (often quoted in milliseconds). Upon removal of the ﬁeld, the corresponding relaxation time is of a comparable scale. The term ER-eﬀect refers to the abrupt change in the apparent viscosity. When the viscosity increases we deal with a positive ER-eﬀect while a decrease in viscosity is called negative EReﬀect [82, 207]. Both the positive and negative ER eﬀect can be enhanced by ultraviolet illumination in some ER systems [83]. This phenomenon is called the photo-electrorheological (PER) eﬀect. Most ERFs are dispersions of polarizable small particles within a nonconducting carrier liquid. The typical range size of the particles entering the structure of an ER-ﬂuid is on the order of 0.10 to 100 µm while the particle volume fraction ranges between 2%–50%. Particles with dimensions below the stated range are liable to execute Brownian motion while larger particles are more liable to sedimentation and also to draw excessive currents. A wide variety of particulate media have been employed in ER suspensions starting from starch, ﬂour, cellulose, ceramic, glass to complex particles such as polyelectrolytes, composite particles (conducting particles coated with a K. Hutter et al.: Electromagnetic Field Matter Interaction in Thermoelastic Solids and Viscous Fluids, Lect. Notes Phys. 710, 279–366 (2006) c Springer-Verlag Berlin Heidelberg 2006 DOI 10.1007/3-540-37240-7 8

280

8 Electrorheological Fluids

thin non-conducting outer layer, doubly coated particles with dielectric cores of high strength and lower mass (see [228]). The impact of the particle shape on ER performance was recently investigated experimentally in [192] by using microspheres and micro-rods as the component of the solid phase of an ER ﬂuid. The dispersing phase of an ER ﬂuid is an insulating oil or other nonconductive liquid. Currently silicone oil, vegetable oil, mineral oil, paraﬃn etc. are used. Besides the suspended particles and the carrier ﬂuid, an ER ﬂuid contains also some additives which could be any polar material that can enhance the ER eﬀect or the stability of the whole suspension. ER ﬂuids that contain a small amount of water are normally called hydrous, in contrast to water-free or anhydrous ER ﬂuids in which no detectable water residue exists. It was demonstrated that the addition of water can enhance the ER-eﬀect. Moreover, the inﬂuence of water in connecting together the particles has been used as the basis of a theory to explain the ER mechanism (the water bridging mechanism). However, a big disadvantage of ER-ﬂuids with moisture content is the limited range of operating temperatures by the freezing and boiling points of water. Fortunately, it was shown that the operating mechanism does not depend on the presence of water and recently, considerable emphasis has been placed on the development of anhydrous particle suspensions. Extensive reviews centred on the material science aspects of ERFs are available [26, 83] and much work continues to be done in order to ﬁnd optimal combinations of material properties (see e.g. [197, 228, 276]). The explanation for the ER eﬀect can be given with the aid of experimental observations at the microscopic level. Under the inﬂuence of an external electric ﬁeld, the initially unordered particles become oriented and attract each other to form particle chains in the ﬂuid along the ﬁeld lines. The chains then aggregate to form columns. These chain-like and columnar structures cause signiﬁcant changes in the resistance to the ﬂow, and the material switches in this way from the liquid state to a solid-like state. In 1949, Winslow [265] reported an ER-eﬀect for certain suspensions and described for the ﬁrst time the phenomenon of induced ﬁbration even though earlier observations on electroviscous eﬀects were reported since 1896 (reviewed in [112]). The basic mechanism for this behaviour is thought to be the ﬁeld-induced particle polarization which is a consequence of the dielectric mismatch between particles and solvent. It should be pointed out that other mechanisms for the ﬁeld-induced increase in viscosity have been suggested including overlap of the diﬀuse counter-ion clouds surrounding neighboring particles [112, 113], electrostatic torque preventing particle rotation in the ﬂow ﬁeld [26], inter-electrode circulation [54] and ﬁeld induced aggregation due to water bridges between particles [208, 222, 223, 229]. A lot of research has been done to develop theoretical models describing these mechanisms and relating the material properties and microscopic phenomena to the measurable macroscopic properties. Most theories are based on the electrostatic polarization mechanism. For an overview of the fundamental physical

8.1 Introduction

281

mechanisms and strategies in relating the microstructural models to the rheological behaviour we refer the reader to the review papers summarizing the main results in this domain [176, 209]. Besides particulate ER suspensions (heterogeneous ERF), there have been developments of homogeneous physical systems which also show dramatic changes in rheological properties upon application of an external electric ﬁeld. Oil-in-oil emulsions and liquid-crystal polymer/oil immiscible systems display a relatively strong ER-eﬀect [102, 110, 111, 199, 245, 272]. This is explained by the increase of domain interactions due to the orientation of elongated molecules. In [66, 167, 168] it is shown that also simple dielectric liquids (insulating oils), of which the viscosity hardly changes when being subject to uniform ﬁelds, can be ER-active when subjected to non-uniform electric ﬁelds (the ER eﬀect is attributed to the electrohydrodynamic convection enhanced by the use of electrodes with ﬂocked fabrics). At last we mention the delicious study of the melt of milk chocolate which also displays ER behaviour [52]. ERFs can be modeled in several diﬀerent ways. ERFs may be analysed by means of molecular dynamic simulations by using diﬀerent models (such as the dipole model, conduction model, equivalent plate conduction model) to establish the equations of motion of the particles. [277, 278]. Another possibility consists in the investigation of their microstructure in order to obtain a macroscopic description of the material [90, 178, 199, 210, 258, 261]. A diﬀerent approach is pursued in the context of continuum mechanics. There are descriptions of ERFs as mixtures of two constituents (the particulate medium and the ﬂuid) [195]. However, many researchers adopted the approach in which ER ﬂuids are treated in a homogenized sense [16, 17, ˇka in [196] and Eckart in [63] for67, 194, 206]. Rajagopal and R˚ uˇ zi c mulated independently governing equations of ER ﬂuids. These formulations have the advantage that they take into account the interactions between the electro-magnetic and the mechanical ﬁelds. After assuming the constitutive law characterizing a certain ﬂuid (see Sect. 8.3 of the present chapter) this approach permits mathematical modeling of the ER behaviour [35, 57, 205]. In our study we also assume ER-ﬂuids to be homogeneous and continuous liquids, and we apply a phenomenological modeling approach in order to predict their macroscopic behaviour. Recently, continuum models were developed which try to reﬂect (at the macroscopic level) ﬁeld induced eﬀects of the micro- and mesostructure of ERFs. In [60] thermodynamic continuum modeling is pursued and the inﬂuence of the ﬁeld generated microstructure is described with the aid of an internal variable theory. We mention the works of Brunn and Abu-Jdayil [33, 34] who carried out a phenomenological study by considering ERFs as ﬂuids with transverse isotropy. This assumption is based on experimental observations according to which ﬁbers are formed upon the application of an electric ﬁeld. In their approach the extra stress tensor of an ERF depends on the rate of strain but also on a vector which characterizes the orientation

282

8 Electrorheological Fluids

and size of the ﬁeld induced ﬁbers. In this way it is possible to describe normal stress eﬀects appearing in viscometric ﬂows. In [65] Eckart and Sadiki applied a polar theory to electrorheological ﬂuids in the context of extended thermodynamics. They succeeded in obtaining a model which accounts for diﬀerent material responses, if the applied electric ﬁeld (assumed to be constant) is either perpendicular or parallel to the ﬂow direction. This fact was expected but the previous models were not capable to reﬂect it. ER-ﬂuids are potentially useful in numerous technical applications. Many of them belong to the automotive industry: shock absorbers, clutches, valves, brakes, dampers, actuators [36, 217]. A good review of the engineering application of ERFs in vibration control can be found in [224]. Non-conventional and advanced actuators may be built using ERFs [75, 139]. Another technological area in which ER ﬂuids oﬀer large promises is virtual reality and telepresence enhanced with haptic (tactile and force) feedback systems [114, 138, 179]. A haptic feedback is a modality for interacting with remote and virtual worlds compared with visual and auditory feedback. ER ﬂuids can be used as smart inks, or to produce photonic crystals or in the polishing industry [83]. Despite the rich research literature about ERFs and the acquired progress in this subject, the application in real-life problems and commercialization of devices based on ERFs have been very limited. The need of high voltage input creates safety problems for the operators especially for the devices that are designed to be in contact with humans. Besides, the problem of their feedback (closed-loop) is diﬃcult to solve because of their complex behaviour. Other obstacles in the development of ERF technologies are related to the composition of ER ﬂuids (e.g. the instabilities caused by the sedimentation tendency of the particles or the limited range of operational temperature). Nevertheless, recent advanced studies led to signiﬁcant improvements in the ﬂuid formulation. It can be foreseen that the large interest concerning these materials and the multitude of research studies focused on their potential applications will ﬁnally improve the capabilities of ER ﬂuids on the one hand and lead to an optimal design for ER devices on the other hand. 8.1.1 Overview The approach described here will be formulated within the framework of continuum mechanics of electromechanical interactions of polarizable materials. In particular, electrorheological ﬂuids are considered to be homogeneous single constituent materials. Following this line, Sect. 8.2 presents the derivation of the governing equations of electrorheology (based on the phenomenological approach developed in [63, 64] with the corresponding jump conditions and the exploitation of the entropy inequality once general constitutive equations for the variables of the problem have been assumed.

8.2 Governing Equations and Constitutive Framework in Electrorheology

283

Section 8.3 treats the constitutive assumptions for the Cauchy stress tensor in greater detail. A review of the constitutive models used to describe ER ﬂuids in the literature is given and two-dimensional constitutive laws, appropriate for numerical simulations originating from the Casson-like and power law models are introduced. The last section deals with applications of the results presented in the previous sections on channel ﬂow of ERF. The boundary value in which the electrodes, ﬂush with the channel, is formulated for both electrically conducting and non-conducting ﬂuids. The simple case of inﬁnite electrodes is analytically solved for the Casson-like model and for the power-law model. Subsection 8.4.3 treats the case of electrodes with ﬁnite length when the electric ﬁeld is inhomogeneous. The equations are non-dimensionalized for both cases, the alternative Casson-like and the alternative power law models. The ﬂow is simulated numerically using a software based on the ﬁnite element method. We study the behaviour of diﬀerent ﬁelds such as velocity, pressure, generalized viscosity and the second invariant of the strain rate tensor near the electrode edges. Comparison with experimental data is performed that validates the simulations. Then we numerically optimize the conﬁguration of the electrodes to obtain an enhancement of the ER-eﬀect. In the last subsection we present shortly the main experimental results as inferred from the literature and obtained with electrodes of diﬀerent geometries (restricted to bidimensional geometries) under a direct current. We, ﬁnally, emphasize here that unlike the general theory dealt with in Chaps. 1 to 6, in which the material response was restricted to thermoelastic behaviour of polarizable and magnetizable bodies, or to magnetoelastic solid bodies, the constitutive behaviour of electrorheological materials includes viscous or plastic eﬀects. This makes the material class somewhat larger than in the earlier chapters, but not signiﬁcantly more complex than before.

8.2 Governing Equations and Constitutive Framework in Electrorheology We use a continuum mechanical model in which electrorheological ﬂuids are considered as homogeneous single constituent materials. Following [63] and essentially [64], we will summarize in this section the main steps of the phenomenological approach, conducted in order to obtain the governing equations of electrorheology with the corresponding jump conditions. The starting point in deriving the system of equations that characterize electrorheological ﬂuids consists in recording the balance laws of thermodynamics of ﬂuids in electromagnetic ﬁelds. We recall here the balances of mass, momentum, moment of momentum and internal energy (2.2.9) and the entropy inequality (2.3.6)

284

8 Electrorheological Fluids

ρ˙ + ρx˙ i,i = 0 , ρ¨ xi = tij,j + ρFie + ρFiext , εijk tkj = 0 , ρU˙ = tij x˙ i,j − qi,i + ρre + ρrext , ρrext , ρη˙ + φi,i ≥ Θ

(8.2.1)

where we assumed that the electromagnetic body couple vanishes. To complete the physical picture we have to add the Maxwell equations which will be considered here in the Maxwell–Minkowski formulation (see Chap. 3, Sect. 3.4), namely (see equations (3.4.10)) ∗

eklm Em,l = − B k , ∗

(8.2.2)

ejlm Hm,l =Dj +Jj ,

(8.2.3)

Di,i = Q ,

(8.2.4)

Bi,i = 0 .

(8.2.5)

We recall the deﬁnitions of the eﬀective electric and magnetic ﬁeld strengths (expressed in terms of the Minkowskian electric and magnetic ﬁeld strengths Em and Hm ) and the conductive current density according to Em := Em + empq x˙ p Bq ,

(8.2.6)

Hm := Hm − empq x˙ p Dq ,

(8.2.7)

Jj := Jj − Qx˙ j .

(8.2.8)

Introducing the electric polarization Pi and the magnetization Mi we have the relations Di := ε0 Ei + Pi , Hi :=

µ−1 0 Bi

− Mi ,

Di := ε0 (Ei − eijk x˙ j Bk ) + Pi , Hi :=

µ−1 0 Bi

(8.2.9)

− ε0 eijk x˙ j Ek − Mi , (8.2.10)

where Mi denotes the eﬀective magnetization Mi := Mi + eijk x˙ j Pk .

(8.2.11)

Relations (8.2.9)1 , (8.2.10)1 and (8.2.11) are identical with relations (3.4.3) and (3.4.4) given in Chap. 3 except for the notation (the superscript M in the expressions for Ei , Hi , Pi , Mi is now dropped).

8.2 Governing Equations and Constitutive Framework in Electrorheology

285

8.2.1 The Electromagnetic Momentum Balance The mathematical expressions for the electromagnetic force and energy supply rate in [63, 64] diﬀer from those treated in Chap. 3. We will now sketch this derivation, but omit details which can be found in [63, 64]. In order to calculate the expression for the electromagnetic force Fie the procedure used in [64] is the following: from the Maxwell–Minkowski equations one may derive an identity which can be interpreted as the electromagnetic momentum balance. Of course, the expression which one may derive in this way contains relativistic terms; since the mechanical balance laws are only non-relativistically valid it follows that one must remove the terms of relativistic order in Fie . This reduction can be sought by non-dimensionalising the equations by introducing adequate scales. In [64] Eckart derives two diﬀerent forms of the electromagnetic momentum balances, which contain diﬀerent expressions for the electromagnetic force. Later on he shows how these diﬀerent expressions for Fie inﬂuence the mechanical quantities. Of course, the velocity obtained from the total momentum balance must – as an observable quantity – be the same for both variants. The ﬁrst variant of the electormagnetic momentum balance is derived by algebraically manipulating the Maxwell-Minkowski equations. Further manipulation with the emerging balance expression then yields the second form of the electomagnetic momentum balance. In [118], Landau and Lifschitz derive the electromagnetic body force from a general expression for the internal energy and deduce an expression for the electromagnetic body force that agrees with the second variant mentioned above. Therefore there are two ways to arrive at these expressions, a formal one from the MaxwellMinkowski equations and a more elegant one from an adequate postulate of the internal energy. Both variants diﬀer in their electromagnetic stress tensors and forces, but not in the quantities (appearing in the MaxwellMinkowski equations), Ej , Dj , Hk , Bk , Jj and Q. These are identical in both forms. In order to derive the electromagnetic momentum balance one starts by postulating the expression for the electromagnetic momentum, viz., gi := eijk ε0 µ0 Ej Hk .

(8.2.12)

From the Maxwell equations (8.2.2)–(8.2.5), and the deﬁnitions (8.2.9)2 and (8.2.10)2 one may then derive the following identity, that can be regarded as the electromagnetic momentum balance g˙i + x˙ l,l gi =

(I) M tij,j

− ρ (I) Fie ,

an identity, in which the electromagnetic stress tensor magnetic force (I) Fie are given by

(8.2.13) (I) M tij

and the electro-

286

8 Electrorheological Fluids (I) M tij

:= Ei Dj + Hi Bj −

ρ (I) Fie :=

1 2 (Ek Dk

+ Hk Bk )δij + ε0 µ0 eilk El Hk x˙ j

− 21 (ε0 ekmn x˙ m Bn Ek − ε0 µ0 ekmn x˙ m En Hk )δij , (8.2.14) Qel Ei + eijk (Dj +Jj )Bk + 12 (Pj Ej,i − Pj,i Ej ) µ0 + (Mk Hk,i − Mk,i Hk ) + eijk Dj Bk −ε0 ejmn x˙ m Bn Ej,i 2

+ε0 µ0 ekmn x˙ m En Hk,i − ε0 µ0

∂ (eijk Ej Hk ) . ∂t

(8.2.15)

This form of the electromagnetic momentum balance is marked with the left superscript “(I)” as an identiﬁer and will subsequently be denoted as the ﬁrst variant. A second form of the electromagnetic momentum balance, marked with the left superscript “(II)” is obtained by diﬀerently writing the divergence of the pressure term in the ﬁrst two lines of (8.2.14) as   −

1 2

  Ek (Dk + ε0 ekmn x˙ m Bn ) + Hk (Bk − ε0 µ0 ekmn x˙ m En ) ,  2 2 34 5 34 5 = ε0 Ek + Pk = µ0 (Hk + Mk ) ,i

and incorporating it in the force term. One then obtains the following electromagnetic momentum balance g˙i + x˙ l,l gi =

(II) M tij,j

where the electromagnetic stress tensor (II) e Fi are given by (II) M tij

− ρ (II) Fie ,

(II) M tij

(8.2.16)

and the electromagnetic force

:= Ei Dj + Hi Bj + ε0 µ0 eilk El Hk x˙ j ,

(8.2.17)

eijk (Dj

ρ (II) Fie := Qel Ei + +Jj )Bk + Dj Ei,j − Bk Hk,i ∂ − ε0 µ0 (eijk Ej Hk ) . ∂t

(8.2.18)

In comparison with the ﬁrst variant the second variant has fewer terms and does not contain the quantities Pj and Mj . We mention that the two electromagnetic momentum balances (8.2.13) and (8.2.16) diﬀer from the electromagnetic momentum balance deﬁned in (2.4.7)1 . However, if we consider gi (2.4.7) = −gi (8.2.13, 8.2.16) , tM ij

(2.4.7) =

(I),(II) M tij

(8.2.19)

(8.2.13, 8.2.16) − gi (8.2.13, 8.2.16)x˙ j , (8.2.20)

we obtain the same form of the electromagnetic momentum equations in both cases.

8.2 Governing Equations and Constitutive Framework in Electrorheology

287

8.2.2 The Electromagnetic Energy Balance In an analogous manner Eckart [64] derives from the Maxwell– Minkowski equations another identity which can be viewed as the electromagnetic energy balance. The starting point of the derivation consists in the choice of the electromagnetic energy. In [64] the expression ω := 12 (Dj Ej + Bj Hj ) ,

(8.2.21)

is used. Analogously, eijk Ej Hk is used as the expression for the Poynting vector. Using then the same equations (8.2.2)–(8.2.5), (8.2.9)2 and (8.2.10)2 and employing similar manipulations as for the derivation of the electromagnetic momentum balance the ﬁrst variant of the energy balance, viz., ω˙ + x˙ l,l ω + (eijk Ej Hk ),i =

(I) M tij x˙ i,j

− ρ (I) re ,

is obtained, where the electromagnetic energy supply

(8.2.22)

(I) e

r is given by

ρ (I) re := 12 (Ej D˙ j − E˙j Dj + B˙j Hj − Bj H˙ j ) + Jj Ej − 12 (ε0 ekmn x˙ m Bn Ek − ε0 µ0 ekmn x˙ m En Hk )x˙ j,j +ε0 µ0 eimn Em Hn x˙ j x˙ i,j .

(8.2.23)

The electromagnetic energy balance 8.2.22 is an identity. Its second variant is described by the equation ω˙ + x˙ l,l ω + (eijk Ej Hk ),i =

(II) M tij x˙ i,j

where the electromagnetic energy supply ρ

(II) e

r :=

− ρ (II) re ,

(8.2.24)

(II) e

r is given by

1 ˙ ˙ ˙ ˙ 2 (Ej Dj − Ej Dj + Bj Hj − Bj Hj ) + Jj Ej 1 − 2 (Ek Dk + Hk Bk )x˙ j,j + ε0 µ0 eimn Em Hn x˙ j x˙ i,j

. (8.2.25)

In order to apply these balances in electrorheology one has to ﬁnd their corresponding non-relativistic approximation. This is done in [64] through a non-dimensionalization process, which we now proceed to explain. 8.2.3 Non-relativistic Approximation Since equations (8.2.13), (8.2.16) and (8.2.22), (8.2.24) contain some relativistic contributions but equations (8.2.1) are valid only in the non-relativistic case, the relativistic terms should be removed. In order to identify them one needs to non-dimensionalize the equations. One can deﬁne the non-dimensional form of most quantities introduced up to this point, if we choose the dimensionless ﬁelds as follows: Ei Ei Di , Ei+ = , Di+ = , E0 E0 ε 0 E0 Pi Ji T = , Ji+ = , ε 0 E0 ε 0 E0

Ei+ = Pi+

(8.2.26)

288

8 Electrorheological Fluids

x˙ + j =

x˙ j xj t , t+ = , , x+ j = V0 L T

(8.2.27)

where the following characteristic quantities were introduced: E0 is a typical ﬁeld strength of the external electric ﬁeld, V0 is a characteristic velocity, L a characteristic length (e.g. the extent of the ﬂow domain) and T is a characteristic time which is in general the inverse value of the electromagnetic frequency. The correct choice of the characteristic quantities of the problem is an important step. This choice determines the domain of validity of the resulting balance laws and consequently of the whole theory. As is evident, the magnetic quantities Bi , Hi , Mi and Mi are missing from the above list. No characteristic magnetic quantity is introduced because, by assumption, there is neither an external magnetic ﬁeld nor a permanent magnetization present.1 So, it seems to be reasonable to build a new characteristic quantity from the above characteristic quantities, that possesses the dimension of a “magnetic unit.”2 In order to do this let us look at the properties of ERFs. Such ﬂuids possess only dielectric and no magnetic properties. In this case we can use the so-called dielectric assumption (see [80], p. 161 and 175), according to which Mk = −ekmn x˙ m Pn .

(8.2.28)

In other words: The magnetization, measured in the rest frame, consists only of a part that appears due to the motion of the electrically polarised medium. This assumption is based on the idea that an observer traveling with the ﬂuid particle experiences no magnetization3 . Using (8.2.11), relation (8.2.28) is tantamount to the statement Mk = 0 .

(8.2.29)

So, a co-moving observer does not observe a measurable eﬀective magnetization. The crucial relation is, however, (8.2.28). We know already how the quantities on the right-side of (8.2.28) have to be non-dimensionalized. From (8.2.26) and (8.2.27) there follows + ekmn x˙ + m Pn = 1

2

3

ekmn x˙ m Pn . V 0 ε 0 E0

It is assumed here that the electrorheological systems studied are free of a permanent magnet (esp. no permanent magnetic electrodes are present). In [205] the approach is diﬀerent. There, a magnetic induction B0 is ﬁrst introduced. Later this is considered small in comparison with the electric ﬁeld strength and it is neglected. This is of course another possibility. But it must be pointed out that there is no a priori knowledge of the magnetic induction. It results as a variable only from the Maxwell-Minkowski equations. One must exclude here that a magnetization due to acceleration eﬀects can appear (remarks related to this see also in [219]).

8.2 Governing Equations and Constitutive Framework in Electrorheology

289

Thus, we may write for the magnetization Mk+ =

Mk , V 0 ε 0 E0

(8.2.30)

in which V0 ε0 E0 may be interpreted as the characteristic magnetization in the electrorheological case. Now, we are able to non-dimensionalize the remaining magnetic quantitites4 . One obtains Bk+ = Hk+

Bk Mk , Mk+ = , V0 ε0 µ0 E0 V 0 ε 0 E0

Hk = , V 0 ε 0 E0

Hk+

(8.2.31)

Hk = , V 0 ε 0 E0

By applying all these transformations the electromagnetic momentum balance in the ﬁrst variant takes the dimensionless form ε0 E02 V02 d gi+ ε0 E02 (I) e,+ ε0 E02 (I) M,+ + + + x ˙ g tij,j − Fi , (8.2.32) = l,l i 2 + Lc dt L L where

d () V0 d ()+ V0 ∂ ()+ + + = := + x ˙ () , Str l ,l dt L dt+ L ∂t+

Str =

L , (8.2.33) T V0

isolates in the bracketed factor the non-dimensionalized material derivative; Str represents the (electric) Strouhal number, and gi and (I) tM ij are given by gi (I) M tij

ε0 E02 V0 + ε0 E02 V0 g := eijk Ej+ Hk+ , i c2 c2 V02 + + + + 2 = ε0 E02 (I) tM,+ := ε E H B 0 0 Ei Dj + ij c2 i j =

−

1 2

−

1 2

Ek+ Dk+

4

(8.2.34)

V02 + + V2 + 2 Hk Bk δij + 02 eilk El+ Hk+ x˙ + j c c

V02 V02 + + + + + + ekmn vm Bn Ek − 2 ekmn vm En Hk δij , (8.2.35) c2 c

More precisely, we require that there is no other magnetic quantity, the value of which clearly exceeds the just introduced quantity. Then (8.2.30) may be interpreted as the highest estimation of the magnetic quantities. If e.g. an electric ﬁeld is used which varies rapidly in time, one should non-dimensionalize the magnetic induction better with ε0 µ0 E0 L/T instead of ε0 µ0 E0 V0 . When one takes into account the electric current one must use a kind of eﬀective conduction instead of ε0 /T . The choice ε0 µ0 E0 V0 is considered an acceptable compromise at the present point.

290

8 Electrorheological Fluids

Fie takes the form ε0 E02 (I) e,+ ε0 E02 V02 + + (I) e + + Fi = Fi := Q Ei + 2 eijk Dj + Str Jj Bk+ ρL ρL c 2 ' ( ( LV0 ∂ ' V + + + eijk Ej+ Hk+ + 02 eijk Dj+ Bk+ + 12 Pj+ Ej,i − Pj,i Ej − 2 ∂t+ c T c V02 V2 + + + + + B E + e x ˙ E H (8.2.36) − 02 ejmn x˙ + kmn m n j,i m n k,i , c c2

and, with the use of (8.2.28),

(I)

in which, the convective derivative of a vector Ai = A0 A+ i has been written as A0 V0 + A0 V0 ∂A+ + + + + + + i Ai = Ai := Str + + x˙ j Ai,j − x˙ i,j Aj + x˙ j,j Ai . L L ∂t (8.2.37) All bracketed terms in (8.2.33)–(8.2.37) are dimensionless. In the ensuing analysis we wish to impose the non-relativistic approximation and shall drop all terms of O(V02 /c2 ), thus assuming V0 c. Beyond this assumption we also request L/T to be of order V0 , so that (LV0 )/(T c2 ) = O(V02 /c2 ) as well. This bounds characteristic lengths not to be too large and characteristic times not to be too small5 . In electrorheological ﬂuids the estimate L/T = O(V0 ) is amply fulﬁlled, so that all above mentioned terms can indeed be neglected. If in the momentum equation all terms of order V02 /c2 are dropped, the ﬁnal form of the non-relativistic electromagnetic momentum balance for electrorheological purposes is given by6 2 V0 V0 L ε0 E02 (1) M,+ (1) e,+ tij,j − Fi + Oi , = 0, (8.2.38) L c2 T c2 (M ),+

where the non-relativistic electromagnetic stress-tensor, (1) tij non-relativistic electromagnetic force (1) Fie,+ are given by (1) M,+ tij (1) e,+ Fi

:= Ei+ Dj+ − :=

Q+ Ei+

+

1 + + 2 Ek Dk δij , ' + + 1 2 Pj Ej,i −

+ + Pj,i Ej

(

and the

(8.2.39) .

(8.2.40)

For the second variant of the electromagnetic momentum balance the same procedure yields 5

6

This would not be reasonable in e.g. astrophysics or high frequency physics. In the ﬁrst case we would have as the characteristic lengths the distances between the planets; in the second case we would have as the characteristic electromagnetic times the inverses to the high frequencies. To distinguish it from the relativistic balances we use now the left superscript “(1)” for the ﬁrst variant instead of “(I)” and analogously the left superscript “(2)” for the second variant instead of “(II)”. From now on the symbol Oi (. . .) (and later also O (. . .)) means “terms of order (. . .)”.

8.2 Governing Equations and Constitutive Framework in Electrorheology

ε0 E02 L

(2) M,+ tij,j

−

(2)

Fie,+ + Oi

V02 V0 L , c2 T c2

= 0, (M ),+

with the non-relativistic electromagnetic stress-tensor, (2) tij relativistic electromagnetic force, (2) Fie,+ given by (2) M,+ tij (2) e,+ Fi

:= Ei+ Dj+ , :=

Q+ Ei+

+

291

(8.2.41) and the non-

(8.2.42) + Dj+ Ei,j

.

(8.2.43)

By applying the same procedure to the electromagnetic energy balances (8.2.22) and (8.2.24) the equation ( ' ε0 E02 V0 d ω + + + + + (I) M,+ + (I) e,+ + x˙ l,l ω − tij x˙ i,j + eijk Ej Hk ,i + r = 0, L dt+ (8.2.44) is obtained for the ﬁrst variant, where V2 ω = ε0 E02 ω + := ε0 E02 12 Dk+ Ek+ + 02 Bk+ Hk+ . (8.2.45) c The electromagnetic energy supply

r

becomes

dEj+ ε0 E02 V0 1 + + Str Jj Ej − Dj+ + r := ρL 2 dt + dDj+ + V02 + dHj+ V02 dBj + V02 + + + + − E + B − H eimn Em Hn x˙ j x˙ i,j + j j j dt+ c2 dt+ c2 dt+ c2 1 V02 V02 + + + + + + + − e x ˙ B E − e x ˙ E H x ˙ . (8.2.46) kmn m n k kmn m n j,j k 2 c2 c2

ε0 E02 V0 r = ρL

(I) e

(I) e,+

(I) e,+

Neglecting in (8.2.45) and (8.2.46) the terms containing the factors V02 /c2 and V0 L/T c2 the following non-relativistic electromagnetic energy balance + ( ' ˜ ε0 E02 V0 d ω + + (m) e,+ + x˙ + ˜ + − (m) tM,+ x˙ + r ij i,j + eijk Ej Hk ,i + l,l ω L dt+ 2 V V0 L +O 02 , 2 = 0 , (m = 1, 2) , (8.2.47) c Tc is obtained where the non-relativistic electromagnetic energy, ω ˜ + , and the (m) e r ) , m = 1, 2, are given non-relativistic electromagnetic energy supply, (ρ by ε0 E02 + + Dk Ek , ˜ + := (8.2.48) ω ˜ = ε0 E02 ω 2

292

8 Electrorheological Fluids

(1) e

r

=

ε0 E02 V0 ρL

(1) e,+

r

ε0 E02 V0 1 := Str Jj+ Ej+ − ρL 2 (2) e

r

=

ε0 E02 V0 ρL

Dj+

dEj+ dDj+ + − E dt+ dt+ j

! ,

(8.2.49)

(2) e,+

r

! dEj+ Ek+ Dk+ x˙ + dDj+ + ε0 E02 V0 1 j,j + + + StrJj Ej − Dj + − + Ej + . (8.2.50) := ρL 2 dt dt 2 In the above relations (8.2.44)–(8.2.50) the bracketed terms, [·], are dimensionless as before. From (8.2.6), there follows 2 V0 V02 + + + + + Em := Em + 2 empq x˙ p Bq = Em + Om , (8.2.51) c c2 and from relations (8.2.9)2 and (8.2.10)2 one obtains, with the help of (8.2.29), Di+ = Ei+ + Pi+ ,

(8.2.52)

+ + Hi+ = Bi+ − eimn vm En .

(8.2.53)

Finally, let us quote the dimensionless form of the Maxwell–Minkowski equations (8.2.2)–(8.2.5). They read E0 V2 + + 02 Bk+ = 0 , (8.2.54) eklm Em,l L c ε 0 E0 V 0 + − Dj+ − Str Jj+ = 0 , (8.2.55) ejlm Hm,l L ε0 E 0 QL + − =0 , (8.2.56) Di,i L ε 0 E0 E0 V 0 + B =0. (8.2.57) Lc2 i,i Now, if one neglects all terms with the factors V02 /c2 and LV0 /(T c2 ) one obtains the electrorheological approximation of the Maxwell-Minkowski equations 2 V0 V0 L E0 + , =0 , (8.2.58) eklm Em,l + Ok L c2 T c2 ' + ( ε 0 E0 V 0 + + + − empq x˙ + E − D =0 , (8.2.59) ejlm Bm p q ,l j − Str Jj L ε 0 E0 QL + − =0 , (8.2.60) Di,i L ε 0 E0 E0 V 0 + B =0. (8.2.61) Lc2 i,i

8.2 Governing Equations and Constitutive Framework in Electrorheology

293

Equation (8.2.58) is crucial. Provided boundary conditions for the electric ﬁeld at domain boundaries are expressible in the electric ﬁeld strength alone, it permits us to calculate the electric ﬁeld decoupled from the magnetic induction and the mechanical ﬁelds. However it is not easy to get rid of the magnetic induction in the electrorheological approximation in all the Maxwell-Minkowski equations. The ﬁeld Bi appears in equations (8.2.54) and (8.2.55) (in Hm ) with diﬀerent factors; this can better be seen in equations (8.2.58) and (8.2.59): the factors in front of the brackets diﬀer through ε0 V0 . Luckily, the magnetic induction can be calculated from equations (8.2.59) and (8.2.61) if the electric ﬁeld and the velocity ﬁeld are known, so it represents a dependent quantity. In the following subsection we will revert the non-dimensionalization and return to dimensional quantities; i.e. we will write e.g. instead of V0 vj+ again vj and so on. Owing to (8.2.51) in the remainder of this chapter we will write Ej instead of Ej . 8.2.4 The Total Balance Laws of Electrorheology Having put the non-relativistic approximation on a rational footing by nondimensionalizing the equations, we may now again return to all approximate equations back in dimensional form. This is often more convenient and certainly physically more transparent. The mass balance (8.2.1)1 remains unchanged. By using (8.2.38) and (8.2.39) the momentum balance equation (8.2.1)2 in the ﬁrst variant takes the non-relativistically correct form ρ¨ xi − (1) tij + Ei Dj − 12 Ek Dk δij − ρFiext = 0 . (8.2.62) ,j

Similarly, with the help of (8.2.41) and (8.2.42) its second variant becomes ρ¨ xi − (2) tij + Ei Dj − ρFiext = 0 . (8.2.63) ,j

Obviously, the Cauchy stress tensor in the ﬁrst and second variants cannot be identical if the emerging models are requested to be the same. This is the reason why they are distinguished in (8.2.62) and (8.2.63) and in the sequel by the superscripts “(1)” and “(2)”. With the aid of (8.2.49) the energy balance (8.2.1)4 for the ﬁrst variant becomes ρ(1) U˙ + (1) qj,j − (1) tij x˙ i,j − ρrext − Jj Ej , + 12 Dj E˙j − D˙ j Ej = 0 , whilst for the second variant, on using (8.2.50), it reads

(8.2.64)

294

8 Electrorheological Fluids

ρ(2) U˙ +(2) qj,j − (2) tij x˙ i,j − ρrext − Jj Ej + 21 Dj E˙j − D˙ j Ej − 12 Ek Dk x˙ j,j = 0 .

(8.2.65)

Here we took into account that, in the two variants, the internal energies and the heat ﬂux vectors can diﬀer from one another. However the energy balances can be written in a more appropriate form. After a simple transformation, by using (8.2.48), we obtain from (8.2.64) ˜˙ + ρ(1) U

(1)

qj,j −

(1)

tij x˙ i,j − ρrext − Jj Ej − Ej D˙ j −

with a modiﬁed internal energy7

(1)

˜ := ρ(1) U + ρ(1) U

1 ˙ j,j 2 Ek Dk x

= 0 (8.2.66)

˜ according to U 1 2 Dk Ek

= ρ(1) U + ω ˜.

(8.2.67)

An analogous change as in (8.2.67) yields instead of (8.2.65) ˜˙ + (2) qj,j − (2) tij x˙ i,j − ρrext − Jj Ej − Ej D˙ j − Ek Dk x˙ j,j = 0 . (8.2.68) ρ(2) U For the derivation of the jump conditions, it is advantageous to use yet another form of the energy balance. What is needed is the conservative form of the total energy balance. To obtain it, let us start from the energy balance (8.2.1)4 ( ' ( d ' ρU + 12 ρx˙ i x˙ i + ρU + 12 ρx˙ i x˙ i x˙ j,j = (tij x˙ i − qj ),j dt ' ( +x˙ i ρFiext + ρFie + ρrext + ρre ,

(8.2.69)

Using (8.2.69), (8.2.38) and (8.2.47) in the dimensional form, respectively, we obtain after straightforward calculations the energy balance for the ﬁrst variant d (1) ρ U + 12 ρx˙ i x˙ i + 12 Ek Dk + ρ(1) U + 12 ρx˙ i x˙ i + 12 Ek Dk x˙ j,j dt + (1) qj + ejik Ei Hk − (1) tij x˙ i − Ei Dj x˙ i + 12 Ek Dk x˙ j ,j

= x˙ i ρFiext + ρrext ,

(8.2.70)

and for the second variant d (2) ρ U + 12 ρx˙ i x˙ i + 12 Ek Dk + ρ(2) U + 12 ρx˙ i x˙ i + 12 Ek Dk x˙ j,j dt + (2) qj + ejik Ei Hk − (2) tij x˙ i − Ei Dj x˙ i = x˙ i ρFiext + ρrext . ,j (8.2.71) 7

The same modiﬁcation was used also in [118].

8.2 Governing Equations and Constitutive Framework in Electrorheology

295

The balances (8.2.70), (8.2.71), now in conservative form, will be used − as mentioned − exclusively to derive the jump conditions for the energy. One may remark that in both equations the magnetic quantity Hk appears. Unlike the energy balances (8.2.66), (8.2.68) the conductive electric current Jj does not appear in (8.2.70), (8.2.71). Thus, eﬀectively one has replaced a constitutive quantity, namely Jj , by another quantity, namely Hk . Concerning the entropy inequality we know or we can relatively easily ¨ller’s entropy principle that for the Maxwell-Minkowski prove with Mu model the entropy ﬂux obeys the Duhem relation (1),(2)

(1),(2)

φj :=

qj

Θ

.

(8.2.72)

Furthermore for both variants the density of the Helmholtz free energy Ψ9(1),(2) is deﬁned in [64] by (1),(2) 9

Ψ :=

(1),(2)

˜ − U

(1),(2)

ηΘ −

1 Dj Ej , ρ

(8.2.73)

where η denotes the entropy as in (8.2.1)5 . ˜ and replace in (8.2.66) If we solve (8.2.73) for the internal energy (1),(2) U ˙ (1),(2) ˜ and (8.2.68) U by the relation obtained from (8.2.73), then new forms of energy balances for the two variants are obtained. These relations contain the external energy supplies (1),(2) rext , which also occur in the entropy imbalance (8.2.1)5 . Eliminating these yields new forms of the entropy inequality in the two variants, which are, respectively, given by −ρ(1) Ψ˜˙ − 12 Dj Ej x˙ i,i − ρ(1) η Θ˙ − Θ,j

(1)

qj − Dj E˙j + Jj Ej Θ

+ (1) tM ˙ i,j ≥ 0 , ij x

(8.2.74)

−ρ(2) Ψ˜˙ − ρ(2) η Θ˙ − Θ,j

(2)

qj − Dj E˙j + Jj Ej Θ

+ (2) tM ˙ i,j ≥ 0 . ij x

(8.2.75)

These relations achieve the fundamental goal of this chapter: the electrorheological entropy inequality for both variants. As one can see, the diﬀerence between both inequalities consists in the factor in front of the velocity divergence. Let us resume now all the total balance equations given so far for both variants. These are (8.2.62), (8.2.66), (8.2.74) and (8.2.63), (8.2.68), (8.2.75), respectively. For comparison of both variants the identiﬁcations (1) (1)

tij =(2) tij + (2)

qj =

(1) 9

(2)

Ψ=

1 2

Dk Ek δij , (1)

(8.2.76) (2)

qj , =⇒ φj = φj , (1) (2) 9 Ψ, U = U , =⇒ (1) η =

(8.2.77) (2)

η.

(8.2.78)

296

8 Electrorheological Fluids

are required if the two variants are to yield identical models. Evidently, the only diﬀerence between the two variants appears in the Cauchy stress tensor. One can compare the results from one variant with those of the other if one takes into account relations (8.2.76)–(8.2.78). Recall also that the electromagnetic quantities Ei , Di , Bi , Hi and Ji are the same in both models. 8.2.5 Jump Conditions In order to complete the electro-mechanical description of electrorheological materials as continuous media, we have to add the jump conditions that must be obeyed by the mechanical and electromagnetic quantities across a discontinuity surface. Let Σ be a smooth surface, not necessarily material (e.g. an inﬁnitely thin wall or a membrane) which separates one part of the body under consideration from another part, and let wi ni be its velocity in the positive direction of the unit normal to Σ. In [64] the easiest case is considered: with the exception of the classical electric quantity Qs (electric surface charge density of the free charges) all surface quantities vanish. In general, the jump conditions for volume balances can be given as (see e.g. [24, 72] and also [161]) Φj + Ξ (x˙ j − wj ) nj = 0 , (8.2.79) (+) (−) (+) (−) in which the abbreviation Aj := Aj − Aj is used where Aj , Aj represent the values of Aj on the immediate exterior (+) and interior (−) side of the surface, and n is the unit normal vector pointing into the positive region of Σ. In addition, Φj denotes the non-convective ﬂux of the volume quantity Ξ and wj is the surface velocity. If Σ is a material surface then we have wj = x˙ j . Let us look now at the physical balances in detail. For the mass balance (8.2.1)1 Ξ = ρ and the non-convective ﬂux, Φj , vanishes. So, (8.2.80) ρ (x˙ j − wj ) nj = 0 . For the total momentum balance (8.2.62) of the second variant we have Ξi = ρx˙ i , Φij = − (2) tij − Ei Dj + 12 Ek Dk δij implying ρx˙ i (x˙ j − wj ) −

(2) M tij

− Ei Dj + 12 Ek Dk δij nj = 0 .

(8.2.81)

The jump condition in the ﬁrst variant looks very similarly, namely ρx˙ i (x˙ j − wj ) − (1) tM (8.2.82) ij − Ei Dj nj = 0 . For the energy balance we start from (8.2.70) and (8.2.71), respectively. With obvious identiﬁcations of Ξ and Φ one obtains

8.2 Governing Equations and Constitutive Framework in Electrorheology

' ( ρU + 12 ρx˙ i x˙ i + 12 Ek Dk (x˙ j − wj ) + qj + ejik Ei Hk (1) 1 − tij + Ei Dj − 2 Ek Dk δij nj = 0 , ' ( ρU + 12 ρx˙ i x˙ i + 12 Ek Dk (x˙ j − wj ) + qj + ejik Ei Hk (2) − tij + Ei Dj nj = 0 .

297

(8.2.83)

(8.2.84)

in the two variants, in which Hk can be eliminated by means of (8.2.53). For the entropy imbalance we start from (8.2.1)5 . For both variants the jump condition is qj ρη (x˙ j − wj ) + (8.2.85) nj ≥ 0 . Θ Finally, the Maxwell-Minkowski equations in the electrorheological approximation, (8.2.58)−(8.2.61), imply (8.2.86) eijk nj Ek = 0 , (8.2.87) eijk nj Hk + eklm (x˙ l − wl ) Dm = 0 , s Dj nj = Q , (8.2.88) (8.2.89) Bj nj = 0 , where Qs represents the electric surface charge density. In what follows we shall exclusively use the second variant. 8.2.6 Discussion The purpose of the ﬁrst part of this section was to motivate the balance equations for electrorheological ﬂuids as they were derived in [64]. Starting from the general balance laws of rational thermodynamics and electrodynamics the system of total balance equations was derived by a consistent specialization to electrorheological behaviour. The total electromechanical balance laws of mass, momentum and the Maxwell equations in the Maxwell–Minkowski formulation were given both in local form and as jump conditions across singular surfaces. The electromagnetic body force and energy supply rates were derived from the Maxwell equations by deducing from these the electromagnetic momentum and energy balances as identities. The production terms in these expressions were identiﬁed as electromagnetic body force and energy supply rate, respectively. However, since the division of the time rates of change of the electromagnetic momentum and energy into ﬂux and production terms are not unique, two variants of electrorheological models were presented, which required connecting relations between energies and stresses, if equivalence of the emerging models are attempted.

298

8 Electrorheological Fluids

The crucial step put forward in this approach was the non-dimensionalization process speciﬁc to the electrodynamic equations. The assumptions that there is no external magnetic ﬁeld and that the magnetic induction is produced exclusively by the motion of the polarized dielectrics, allowed us to neglect the magnetic terms compared with the electric terms in most (but not all) of the electrodynamical equations and, consequently, also the total balances8 . The essential results are the simpliﬁed balance equations of thermomechanics and electrodynamics which will be used in the remainder of this work; this is why we put them here together. These equations are (in due order) the mass balance, the linear momentum balance, the balance of moment of momentum, the energy balance, the entropy inequality as well as the four Maxwell-Minkowski equations in the electrorheological approximations; explicitly, ρ˙ + ρ x˙ j,j = 0 , ρx¨i − (2) tij + Ei Dj

(8.2.1)1 ,j

−

ρFiext

= 0,

eijl (2) tlj = 0 , ˜˙ + qj,j − (2) tij x˙ i,j − ρrext − Jj Ej − Ej D˙ j ρU −Ek Dk x˙ j,j = 0 , qj ˙ (ERF) − ρψ˜ − η Θ˙ − θ,j − Dj E˙j + Jj Ej + (2) tij x˙ i,j ≥ 0 , Θ eklm Em,l = 0 , ∂Dj − Jj , ejlm µ−1 0 Bm + empq vp (Dq − ε0 Eq ) ,l − ∂t − Qx˙ j = 0 , Di,i − Q = 0 , Bi,i = 0 .

(8.2.63) (8.2.1)3 (8.2.68) (8.2.75) (8.2.90)

(8.2.91) (8.2.4) (8.2.5)

Henceforth they will be labeled and called ERF equations. 8.2.7 Constitutive Equations The balance equations in the electrorheological approximation given above are not suﬃcient to determine all the unknowns of the problem; so additional relations reﬂecting the speciﬁc properties of the studied material must be postulated. The starting point of this procedure consists of the choice of the 8

Even if the magnetic induction would be produced primarily by a time changing electric ﬁeld or by an electric current, the neglect of the magnetic compared to the electric quantities is under practically relevant circumstances still justiﬁed. In general, the situation changes only drastically when an external magnetic ﬁeld is present.

8.2 Governing Equations and Constitutive Framework in Electrorheology

299

independent ﬁelds from all the physical variables involved in the governing equations. Then we must establish the appropriate constitutive equations for the remaining variables. In [64], by obeying the principle of material frame indiﬀerence, the independent variables have been chosen as ρ , Θ , Ej , dij ,

(8.2.92)

where dij denotes the rate of the strain tensor dij := 12 (x˙ i,j + x˙ j,i ) ,

(8.2.93)

which is objective. The presence of this quantity among the independent variables is motivated by the fact that ER-materials behave in general as viscous ﬂuids. That Θ,i is not among the variables (8.2.92) implies that eﬀects of heat conduction are not considered. Its incorporation would be formally quite obvious and easy, but we omit such a dependence, because in subsequent applications it will not be considered. It is important to note that the magnetic ﬂux density Bi is not an independent constitutive variable. Such an assumption is reasonable, because Bi is likely very small, since according to (8.2.28) magnetization is induced only by polarization. On this basis Bi can then be computed from (8.2.91) and (8.2.5) if the electric ﬁeld strength and the velocity are known. It then follows that Bi will not inﬂuence the mechanical equations, and it will neither intervene in the electrical problem as will soon be seen. Although the electric ﬁeld may also be space-dependent, in a ﬁrst approximation its gradient will not be considered an independent constitutive variable either. The dependent constitutive quantities of the problem are C := {tij , Dj , Jj , U, Ψ˜ , qj } ,

(8.2.94)

where we have dropped the left upper index “(2)” from tij , and they are of the form (8.2.95) C = C(ρ, Θ, Ei , dij ) . Henceforth we shall focus attention on formulation “(2)”. Evaluation of the entropy inequality Evaluation of the entropy inequality will bring the constitutive equations to their ﬁnal form. Substituting the constitutive function for the Helmholtz free energy given in (8.2.94) and (8.2.95) into (8.2.75) and performing the time diﬀerentiation according to the chain rule results in

300

8 Electrorheological Fluids

∂ Ψ˜ −ρ η + ∂Θ

Θ˙ − ˜

2 ∂Ψ

+ tij + ρ

∂ρ

∂ Ψ˜ Dj + ρ ∂Ej

qj ∂ Ψ˜ ˙ E˙j − ρ dkl − Θ,j ∂dkl Θ

! δij dij + Jj Ej ≥ 0 .

(8.2.96)

Exploitation of this inequality follows the same procedure outlined earlier in Chap. 3. The inequality is explicitly linear in Θ˙ , E˙j , d˙kl , Θ,j .

(8.2.97)

Since in admissible thermodynamic processes all these terms may have any arbitrarily assigned values, it follows that each of the coeﬃcients of these variables in 8.2.96 must be identically zero. This implies the relations ∂ Ψ˜ , ∂Θ ∂ Ψ˜ Dj = −ρ , ∂Ej ∂ Ψ˜ 0=− , ∂dkl qj =0, Θ η=−

(8.2.98) (8.2.99) (8.2.100) (8.2.101)

and the residual inequality ˜

2 ∂Ψ

tij + ρ

∂ρ

! δij dij + Jj Ej ≥ 0 .

(8.2.102)

Cross-diﬀerentiating (8.2.98)–(8.2.100) we obtain ∂η =0, ∂dkl ∂Dj =0, ∂dkl ∂η ∂Dj ρ . = ∂Ej ∂Θ

(8.2.103) (8.2.104) (8.2.105)

From (8.2.103), (8.2.104) it follows that the entropy and the electric displacement cannot depend on the rate of strain tensor. Alternatively, (8.2.105) establishes a connection between the constitutive laws for η and Dj ; so not all the constitutive assumptions can be chosen independently. If we use the deﬁnitions of the thermodynamic pressure and if we decompose the Cauchy stress tensor into a spherical tensor containing the dynamic pressure −pδij and an extra stress tensor teij , viz,

8.2 Governing Equations and Constitutive Framework in Electrorheology

∂ Ψ˜ , ∂ρ teij := tij + pδij ,

p := ρ2

301

(8.2.106) (8.2.107)

the imbalance (8.2.102) can be written as ργ := teij dij + Jj Ej ≥ 0 ,

(8.2.108)

where γ is the density of entropy production. An important step is now the satisfaction of (8.2.108). A necessary condition to fulﬁll the thermodynamic equilibrium requirement, γ = 0 (no entropy is produced), is & & ∂γ && ∂γ && = 0, = 0 , (8.2.109) ∂dA &E ∂Ep &E where we denoted by dA the 6 independent components of the rate of strain tensor (8.2.110) (dA ) := (d11 , d12 , d13 , d22 , d23 , d33 ) . In (8.2.109) and in the sequel the index “E” will denote thermodynamic equilibrium. Further, one has to treat separately the case of electric conducting ﬂuids (namely Jj = 0) and the case of non-conducting ﬂuids (namely Jj = 0) because the results of this evaluation will be diﬀerent in the two cases. 1. Electrically conducting ﬂuids. In equilibrium it follows from (8.2.108) that the independent variables dpq and Ep must vanish, dpq |E = 0 ,

and

Ep |E = 0 .

(8.2.111)

If we replace in (8.2.109) γ by the formula given on the left-hand side of (8.2.108) and take into account that teij = teij (ρ, Θ, Ek , dkl ) as well as Jj = Jj (ρ, Θ, Ek , dkl ), then (8.2.111) yields & & e & ∂ tij ∂dij ∂dij && ∂Jj dij + teij + Ej && = teij = 0 , (8.2.112) ∂dA ∂dA ∂dA ∂dA &E E and

& & ∂ teij ∂Jj dij + Ej + Jp && = Jp |E = 0 . ∂Ep ∂Ep E

(8.2.113)

For the conductive current Jj , (8.2.113) implies when dpq = 0

and Ep = 0 ,

then

Jj = 0 .

(8.2.114)

In view of the choice of the independent variables (8.2.92) this condition is always fulﬁlled. From (8.2.112), by taking into account the symmetry of the extra stress, it follows that teij cannot be composed of terms containing only ρ and Θ. Consequently,

302

8 Electrorheological Fluids

when

dpq = 0

then teij = 0 .

and Ep = 0 ,

(8.2.115)

2. Electrically non-conducting ﬂuids. Unlike for conducting ﬂuids the electric ﬁeld does not need to vanish in thermodynamic equilibrium. From (8.2.109)1 there follows a condition identical to (8.2.112), whilst from (8.2.109)2 one obtains a condition which is always fulﬁlled if ∂teij /∂Ep |E is bounded which we will assume. However, condition (8.2.112) is stronger in this case: teij must vanish whenever dmn is zero and consequently teij cannot contain terms formed only with ρ, Θ and El . A suﬃcient condition guaranteeing that (8.2.108) is fulﬁlled at thermodynamic equilibrium is that the matrix of the second-order derivatives (with respect to dA and Ei ) must be positive semideﬁnite (see [161]). More precisely, this means that all the principal subdeterminants of the matrix  ∂ 2 γ ∂ 2 γ & & & & ∂Ep ∂Er ∂Ep ∂dA (3 × 3) (3 × 6) && &  = (8.2.116)   && 2 2 (6 × 3) (6 × 6) &E ∂ γ ∂ γ & ∂d ∂E ∂d ∂d A

r

A

B

E

must be larger or equal to zero. It is diﬃcult to evaluate these conditions in general, but we will do it later for concrete constitutive assumptions. Constitutive functions for the electrical quantities Let us discuss now the constitutive laws for the dependent electrical quantities. Starting from (8.2.94) and (8.2.95), Eckart considers in [64] the most general expressions for isotropic polar vector functions (here Dj and Jj ) of a polar vector (here Ej ) and a symmetric tensor of second order (here djk ). These representations are9 Dj = εEj ,

(8.2.117)

Jj = σ1 Ej + σ2 djk Ek + σ3 dj dk Ek .

(8.2.118)

(see e.g. [72] and [220]). The quantity ε denotes the eﬀective permittivity of the ER-material at hand and σ1 , σ2 , σ3 represent the eﬀective conductivities. In view of (8.2.104) one can easily see that ε cannot depend on the strain rate tensor dij ; therefore ε = ε (ρ, Θ, Ek Ek ) .

(8.2.119)

Alternatively the eﬀective conductivities can be any functions of the invariants of the independent variables 9

In (8.2.117) two additional terms ε2 djk Ek and ε3 djl dlk Ek are missing because of (8.2.104)

8.3 Constitutive Laws for the Cauchy Stress Tensor

303

σA = σA (ρ, Θ, Ek Ek , dkk , djk dkj , djk dkl dlj , Ej djk Ek , Ej djk dkl El ) , (8.2.120) where A = 1, 2, 3. This is, of course very complicated, and in practice one will restrict the number of independent variables to just a few, e.g. σA = σA (ρ, Θ, Ek Ek , djk dkj ) .

(8.2.121)

The electrical model of order zero The most simple meaningful constitutive assumptions for the electrical quantities are models for which ε and σA (A = 1, 2, 3) do not depend on the rate of strain tensor [they are of “zeroth order” in dij 10 (ε is of course always of zeroth order in dij according to (8.2.119))]. Moreover, the constitutive parameters must also be independent of the electric ﬁeld: we consider here linear dielectrics and Ohmian conductors. With these restrictions one obtains from (8.2.119) and (8.2.121) ε = ε (ρ, Θ) , σ1 = σ100 (ρ, Θ) , σ2 = σ3 = 0 .

(8.2.122)

So, the electrical constitutive parameters are constant for constant density and temperature. All these assumptions agree with the experimental results presented by Eckart [64] who quotes a relevant selection from the measurements done by the Rheology work-group, Department of Fluid Mechanics of the University of Elangen-N¨ urenberg, see [2, 269, 270]. In our further approach we will adopt the material description given by (8.2.122). A discussion about the constitutive function of the Cauchy extra stress tensor teij is postponed until the next section where a literature review of the constitutive models for teij used to describe ER ﬂuids is given.

8.3 Constitutive Laws for the Cauchy Stress Tensor 8.3.1 Models Proposed in the Literature A key step in electrorheology is to relate the theory with practical applications namely with the results from measurements and computations. Usually, the theoretical approaches are very abstract, general and diﬃcult to use in concrete situations whereas the empirical approaches are applicable but often too particular. In the electrorheological ﬁeld noticeable eﬀorts are made 10

Measurements (see especially [2]) show that the measurable electric current can depend on the shear rate. This dependence is, however, largely inﬂuenced by the temperature, frequency of the electric ﬁeld and mostly by the composition of the ERF. For the ERF Rheobay the shear rate dependence is, however, small; in this work this dependence will be neglected.

304

8 Electrorheological Fluids

from both sides to describe in a better and more accurate way the response of electrorheological ﬂuids to external ﬁelds. We will review brieﬂy the most important theoretical models for the expression of the Cauchy stress tensor according to (8.2.94) and (8.2.95) that have so far been proposed in the literature. While these proposals are three-dimensional expressions for the Cauchy stress, in most applications one-dimensional models are used. If one deals with numerical computations involving two or three-dimensional models one can either generalize the one-dimensional models or choose particular forms of the general models taking into account also the experimental characterization of the ERFs. The most general form of the constitutive function for the Cauchy stress tensor that depends on the objective independent variables ρ, Θ, dij and Ei (see (8.2.94) and (8.2.95) is given by (see [220]) tij = −pδij + teij = (−p + α1 )δij + α2 Ei Ej + α3 dij + α4 dik dkj +α5 (Ei djk Ek + dik Ek Ej ) + α6 (Ei djk dkl El + dik dkl El Ej ) , (8.3.1) where αi , i = 1, . . . , 6 are functions of the invariants ρ , Θ , Ek Ek , dkk , djk dkj , djk dkl dlj , Ej djk Ek , Ej djk dkl El .

(8.3.2)

This general constitutive law was ﬁrst proposed by Rajagopal and Wineman in [194]. However, they treated the electric ﬁeld as a constant when calculating the velocity ﬁeld for the ﬂow problems formulated in their paper. In [196] two special cases of (8.3.1) are discussed. In the ﬁrst it is assumed that the stress is linear in dij and quadratic in Ei and hence the material parameters have the form α1 α2 α3 α4 α5 α6

= α11 + α12 dkk + α13 Ek Ek + α14 Ek Ek djj + α15 Ej djk Ek , = α21 + α22 dkk , = α31 + α32 Ek Ek , =0, = α51 , =0 ,

(8.3.3)

where αij are functions of ρ and Θ only. The subcases of (i) a compressible, (ii) a mechanically incompressible but electrically compressible and (iii) an incompressible ﬂuid are considered and for each of them the restrictions imposed on tij by the Clausius–Duhem inequality are given. The second case pertains to the non-linear model of incompressible ERFs with shear dependent viscosities. First it is assumed that α4 = α6 = 0 .

(8.3.4)

The choices for the material parameters α2 , α3 and α5 reﬂect a combination of a Newtonian and power-law like behaviour where the power exponent11 11

We adopted a diﬀerent notation than in [196], where p is used instead of n, to avoid confusion with the pressure p

8.3 Constitutive Laws for the Cauchy Stress Tensor

305

can be a function of Ek Ek . Concretely, it is assumed that α2 = α20 + α21 (dlm dml )(n−1)/2 , α3 = α30 + α31 (dlm dml )(n−2)/2 + α32 Ek Ek +α33 Ek Ek (dlm dml )(n−2)/2 , (n−2)/2

α5 = α50 + α51 (dlm dml )

(8.3.5)

,

where αij are functions of Θ. The material function n depends on Ek Ek and satisﬁes (8.3.6) 1 < n∞ ≤ n(Ek Ek ) ≤ n0 < ∞ , where n0 =

lim

Ek Ek →0

n(Ek Ek ) ,

n∞ =

lim

Ek Ek →∞

n(Ek Ek ) .

(8.3.7)

An alternative model to (8.3.5) is also given; in this model (dlm dml )β/2 is replaced by (8.3.8) (ξ + dlm dml )β/2 or (ξ + (dlm dml )1/2 )β , where β = n − 1 or n − 2. The purpose of the addition12 of ξ in the representation 8.3.8 prevents the model for n ∈ (1, 2) from developing inﬁnite shear viscosity at zero stretching and a yield-like behaviour as in the ﬁrst model (8.3.5). Simpliﬁed models of (8.3.5) and (8.3.8) corresponding to speciﬁc electrorheological ﬂuids (n ≡ 2 or n ≡ 2) are considered and restrictions for the corresponding coeﬃcients αij are obtained. Finally, a model which includes all the discussed approximating models except that with n ∈ (1, 2) and α30 = α32 = α50 = 0 is proposed as follows: tij = −pδij + α21 ((ξ + dlm dml )(n−1)/2 − 1)Ei Ej +(α31 + α33 Ek Ek )(ξ + dlm dml )(n−2)/2 dij +α51 (ξ + dlm dml )(n−2)/2 (Ei djk Ek + dik Ek Ej ) ,

(8.3.9)

where n satisﬁes (8.3.6). All cases formulated and discussed in [196] are ˇka studied in detail mathamenable to mathematical analysis. In [205], R˚ uˇ zi c ematical issues such as existence, uniqueness and stability of weak and strong solutions for steady ﬂow of incompressible shear dependent electrorheological ﬂuids with the stress given by (8.3.9). In view of the dependence of the material function n on the magnitude of the electric ﬁeld, this problem is described by an elliptic or parabolic system of partial diﬀerential equations (PDE) exhibiting so-called non-standard growth conditions, i.e. the elliptic operator teij = tij + pδij satisﬁes the inequalities 12

Here ξ is a quantity having the dimension 1/s2 in (8.3.8)1 and 1/s in (8.3.8)2 which for the sake of the mathematical treatment is taken to be equal to 1 s−2 (s−1 ). The choice ξ = 1 is justiﬁed, because the role of the extra constant ξ is to regularize the system of partial diﬀerential equations at zero stretching

306

8 Electrorheological Fluids

teij (d, E)dij ≥ c0 (1 + Ek Ek )(ξ + dlm dml ) teij (d, E) teij (d, E) ≤ c1 (ξ + dlm dml )

n0 −1 2

n∞ −2 2

dlm dml ,

Ek Ek .

(8.3.10) (8.3.11)

In [205] the case of unsteady ﬂows of shear dependent ERFs is also treated for the constitutive function tij = −pδij + α31 (1 + Ek Ek )(ξ + dlm dml )(n−2)/2 dij ,

(8.3.12)

where α31 > 0 and n = n(Ek Ek ) satisﬁes (8.3.6) with n∞ ≥ 2. The existence of weak and strong solutions global in time for large data under certain restrictions on n∞ and n0 and the uniqueness of the strong solution are proved. The model proposed by Eckart in [63]13 consists of an extension of the model (8.3.5) modiﬁed as in (8.3.8)1 , tij = −pδij + [α20 + α21 Z −a+1/2 + α22 Z −b+1/2 ]Ei Ej +[α30 + α31 Z −a + α32 Z −b ]dij +[α40 + α41 Z −a−1/2 + α42 Z −b−1/2 ]dik dkj +[α50 + α51 Z −a + α52 Z −b ](Ei djk Ek + dik Ek Ej ) +[α60 + α61 Z −a−1/2 + α62 Z −b−1/2 ](Ei djk dkl El + dik dkl El Ej ) , (8.3.13) where Z := D02 + dmn dnm . Here αij as well as the exponents a and b are material parameters that can depend only on ρ, Θ and Ek Ek , while D0 denotes a constant reference shear rate. The reasons for this choice of the constitutive function are explained in detail in [63] (see also [64]). We mention only some selected arguments. First, this model includes the Casson model as a special case, to be presented below in (8.3.49)–(8.3.50) and quite successfully used (in one-dimensional form) to ﬁt measured data for one of the ERFs, tested in [2]. Second, by introducing coeﬃcient expansions in Z with two exponents a and b, it is possible to describe the measurements with constant exponents. 13

The same author presents in [64] a more detailed approach where he treats also the so-called mechanical model of order 1 (the stress is linear in dij and quadratic in Ei ) which was discussed also in [196] as mentioned above. Unlike Rajagopal ˇka, Eckart considers only incompressible ﬂuids but in addition to and R˚ uˇ zic the results presented in [196] he gives the consequences of the entropy inequality on this model also for electrically conducting ﬂuids. The particularization of this model on a viscometric ﬂow under uniform electric ﬁeld perpendicular to the ﬂow direction shows Newtonian behaviour with a viscosity depending on the electric ﬁeld. This behaviour is not supported by the experimental data where one observes strongly non-linear dependence of the shear stress on shear rate (for more details see [64]).

8.3 Constitutive Laws for the Cauchy Stress Tensor

307

[Having only one exponent that does not depend on the electric ﬁeld, it is in general not possible to ﬁt the data]. This is advantageous for inhomogeneous electric ﬁelds (for which Ek depends on xj ) since we then avoid the exponents to be dependent on the coordinate xj . The quantity D02 , which in [196] has been chosen as D02 = 1, exhibits an important meaning. A non-vanishing D0 plays the same role as ξ in (8.3.8). Speciﬁcally, it prevents the model for certain choices of the exponents a and b from developing an inﬁnite viscosity limit at vanishing stretching. For numerical calculations a non-vanishing value D0 = 0 avoids singularities that may otherwise cause problems in the computations. For some particular cases of (8.3.13), however, e.g. in viscometric ﬂows (see (8.3.27)–(8.3.30), (8.3.33)– (8.3.36)), the choice D0 = 0 allows calculation of analytical solutions for the velocity (see Subsect. 8.4.2). However, when an inhomogeneous electric ﬁeld is considered, the ﬂow is no longer viscometric and numerical solutions must be sought by choosing certain positive values for D0 . The phenomenological approach was continued in [63] (see also [64]) with the investigation of model (8.3.13) in a viscometric ﬂow with a constant electric ﬁeld perpendicular to the ﬂow direction (e.g. shear ﬂow in a plane channel under an electric ﬁeld produced by two inﬁnite electrodes placed along the channel walls – see Subsect. 8.4.2), i.e., E1 = E3 = 0 , d12 = d21 =

1 2

V , h =: 12 γ˙ , dij = 0 otherwise .

E2 = − x˙ 1,2

(8.3.14) (8.3.15)

Then, it is straightforward to show that Z = D02 + 12 γ˙ 2 , τ := t12 = t21 = (β¯0 + β¯1 Z

(8.3.16) −a

+ β¯2 Z

−b

)γ˙ ,

(8.3.17)

t13 = t31 = 0 , t23 = t32 = 0 ,

(8.3.18) (8.3.19)

t11 = −p + 14 (α40 + α41 Z −a−1/2 + α42 Z −b−1/2 )γ˙ 2 ,

(8.3.20)

V2 = −p + (α20 + α21 Z −a+1/2 + α22 Z −b+1/2 ) 2 h +(ψ¯0 + ψ¯1 Z −a−1/2 + ψ¯2 Z −b−1/2 )γ˙ 2 , = −p ,

(8.3.21) (8.3.22)

t22

t33

N1 := t11 − t22 = −(α20 + α21 Z −a+1/2 + α22 Z −b+1/2 ) − 12 (α60 + α61 Z −a−1/2 + α62 Z −b−1/2 )

V2 2 γ˙ , h2

N2 := t22 − t33 = (α20 + α21 Z −a+1/2 + α22 Z −b+1/2 ) +(ψ¯0 + ψ¯1 Z −a−1/2 + ψ¯2 Z −b−1/2 )γ˙ 2 .

V2 h2 (8.3.23) 2

V h2

(8.3.24)

308

8 Electrorheological Fluids

where β¯i := 12 ( α3i + α5i V 2 /h2 ) , i = 0, 1, 2 , ψ¯i := 14 ( α4i + 2α6i V 2 /h2 ) , i = 0, 1, 2 ,

(8.3.25) (8.3.26)

and αij = αij (ρ, Θ, V 2 /h2 ). Unfortunately there are no measurements available for the normal stresses t11 , t22 , t33 . That is why the parameters α4i and α6i , i = 0, 1, 2, which are primarily responsible for non-vanishing values of N1 and N2 , will be set equal to zero in the constitutive models used in the numerical computations described below, see (8.3.39), (8.3.65), (8.3.66). We also mention that N1 = 0 when the electric ﬁeld vanishes which is an unrealistic property, but for an ERF this is acceptable since these ﬂuids exhibit no normal stress eﬀects in the absence of the electric ﬁeld. For certain choices of D0 , a, b and β¯i the shear stress formula (8.3.17) agrees with the corresponding formula that may also be derived from model (8.3.8) if that is restricted to the viscometric ﬂow in question. Model (8.3.17) with D0 = 0 includes also the most popular models14 used in a onedimensional form in electrorheology: the Bingham model, the Casson model and the power-law model [2, 176, 258, 269, 270]. Let us employ the upper indices B, C and P to denote the quantities corresponding to each of them. • By choosing β¯0 = η B , a = 12 , β¯1 = model15 is found

√1 τ B 2 y

τ B = τyB + η B γ˙ , τ

B

=

−τyB

B

+ η γ˙ ,

and β¯2 = 0 the linear Bingham

τ B > τyB ,

γ˙ > 0 ,

(8.3.27)

−τyB

, γ˙ < 0 ,

(8.3.28)

τ

B

<

where τyB ≥ 0 is the yield stress and η B > 0 the viscosity. • The non-linear Casson-model possesses also two material parameters: the yield stress τyC ≥ 0 and the viscosity η C > 0. Its equation reads τ C = τyC + 2 (τyC η C γ) ˙ 1/2 + η C γ˙ ,

τ C > τyC ,

τ C = −τyC − 2 (τyC η C (−γ)) ˙ 1/2 + η C γ˙ ,

τ C < −τyC , γ˙ < 0 , (8.3.30)

γ˙ > 0 , (8.3.29)

and is obtained by choosing the parameters in (8.3.17) as follows β¯0 = η C , a = 14

15

1 2

, β¯1 =

√1 τ C 2 y

, b=

1 4

, β¯2 = 23/4 (η C τyC ) .

(8.3.31)

For models with yield stress one can deduce only the equations describing the liquid-like behaviour. In [63], equations (8.3.27), (8.3.29), (8.3.33) and (8.3.35) are said to be valid also for γ˙ = 0. We agree with this only in the sense of a subsequent extension by continuation and not as a deduction from the model (8.3.17) since when a > 0, Z is not deﬁned for γ˙ = 0 (except when n > 1 for the power law model which is valid for γ˙ = 0). Equations (8.3.28), (8.3.30) and (8.3.34) are not given in [63]; these can also be deduced from (8.3.17) and we give them here for completeness.

8.3 Constitutive Laws for the Cauchy Stress Tensor

309

• The power-law model is also non-linear but it does not possess a yield region and thus continuously connects a zero stretching regime with any such non-zero regime. For its parameterization one must choose β¯0 = 0 , a = 12 (1 − n) , β¯1 = 2−(1−n)/2 m , β¯2 = 0 ,

(8.3.32)

where m > 0 and n > 0 are the two model parameters and (8.3.17) becomes τ P = mγ˙ n , τ

P

γ˙ > 0 ,

(8.3.33)

= −m(−γ) ˙ , γ˙ < 0 .

(8.3.34)

n

Then, by analysing the experimental data measured for the electrorheological ﬂuid Rheobay TP AI 3565, the Casson-like model, introduced by choosing D0 = 0, a = 12 and b = 14 , is found to be a very suitable model (at least for the ﬂuid Rheobay) τ Cl = (η0 + β0 )γ˙ + 21/2 β1 + 21/4 β2 γ˙ 1/2 , τ Cl > 21/2 β1 ≥ 0 , γ˙ > 0 , τ

Cl

= (η0 + β0 )γ˙ − 2

1/2

β1 − 2

1/4

1/2

β2 (−γ) ˙

(8.3.35)

,

τ Cl < −21/2 β1 ≤ 0 , γ˙ < 0 ,

(8.3.36)

where the upper index “Cl” is used to denote the shear stress for the Cassonlike model. Here, β1 = β¯1 , β2 = β¯2 and β0 = β¯0 − η0 must obey the relation βi (V = 0) = 0 ,

i = 0, 1, 2 ,

(8.3.37)

where η0 is the dynamic viscosity in the absence of an electric ﬁeld. By imposing (8.3.37) it is demanded that the ﬂuid has Newtonian behaviour at vanishing electric ﬁeld. Whereas in the Casson model, the coeﬃcient of γ˙ 1/2 is connected to the other material coeﬃcients, in the model (8.3.35)– (8.3.36) it is independent of these. Consequently, the Casson-like model can be particularized to the Bingham model. The values of η0 and β1 , β2 may diﬀer for diﬀerent electric ﬁelds. A table with values found by ﬁtting the data from the measurements obtained in a rotational viscometer for the ER-ﬂuid Rheobay at diﬀerent electric ﬁelds is given in Subsect. 8.4.3 (Table 8.1). Finally, taking into account the experimental results, we now impose additional, new assumptions in the general model (8.3.13) to obtain a simpler constitutive function which is still able both to describe the measurements and to make simpler analytical solutions possible. First, the parameters responsible for the normal stress eﬀects, (see (8.3.23), (8.3.24), are neglected16 , 16

Actually, the parameter α2 can also be responsible for non-trivial normal stress diﬀerences. However, it is almost impossible to describe realistically the normal stress eﬀects only with this parameter.

310

8 Electrorheological Fluids

α40 = α41 = α42 = α60 = α61 = α62 = 0 , and, second, the values a =

1 2

and b =

1 4

(8.3.38)

are chosen. This leads to

tij = −pδij + [α20 + α21 + α22 (D02 + dmn dnm )1/4 ]Ei Ej + 2η0 dij +[α30 + α31 (D02 + dmn dnm )−1/2 + α32 (D02 + dmn dnm )−1/4 ]dij +[α50 + α51 (D02 + dmn dnm )−1/2 + α52 (D02 + dmn dnm )−1/4 ] [Ei djk Ek + dik Ek Ej ] .

(8.3.39)

In [63] this is called the extended Casson model. When the electric ﬁeld vanishes, the factors α30 , α31 and α32 must equally vanish to guarantee Newtonian behaviour for vanishing electric ﬁeld. The entropy inequality imposes restrictions on the material parameters. The necessary condition (8.2.114) is always fulﬁlled due to the choice of the independent variables. On the other hand, (8.2.115) is identically fulﬁlled for conducting ﬂuids while for non-conducting ﬂuids it requests that 1/2

α20 + α21 + α22 D0

=0,

(8.3.40)

The fact that the dependence of the parameters αij on the electric ﬁeld is unknown makes evaluation of the suﬃcient condition (8.2.116) diﬃcult. For non-conducting ﬂuids the following relations can be deduced: η0 ≥ 0 ,

(8.3.41) −1/2

2η0 + [α30 + α31 D0−1 + α32 D0 +2[α50 + α51 D0−1 +

−1/2

2η0 + [α30 + α31 D0−1 + α32 D0 +[α50 + α51 D0−1 +

]

−1/2 α52 D0 ]E12

≥0,

(8.3.42)

+ E22 ) ≥ 0 .

(8.3.43)

]

−1/2 α52 D0 ](E12

We regard the two approaches presented in [63] and [196] presently as the most advanced single constituent constitutive proposals for the Cauchy stress and electric ﬁeld. They are therefore important and relevant for our treatment of particular problems to be attacked below. Nevertheless, in order to provide a broader view of attempts to describe ER-ﬂuids, we consider it worthwhile to mention also other constitutive models introduced in the speciﬁc literature. We continue by mentioning two studies which are interesting, to a greater extent from the mathematical point of view. Both of them treat extensions of the Bingham model. In [67] the authors propose a so-called extension of the Bingham model determined in terms of the minimization of the global dissipation energy. The Cauchy stress (in tensorial form) is given by t = −pI + γ

|E| (dE ⊗ E + E ⊗ dE) + η d . |dE|

(8.3.44)

8.3 Constitutive Laws for the Cauchy Stress Tensor

311

Obviously, it is not well deﬁned when dE = 0, which characterizes the “rigid zones”. It can be proved that the shear stress has to exceed the threshold γ|E|2 outside the rigid zones; so this quantity may be viewed as an equivalent of the yield limit of the standard Bingham model. However, the stress tensor is not used directly to formulate the boundary value problem to be solved. The velocity ﬁeld is computed as the solution of a non-smooth minimization problem for the global energy dissipation. This minimization problem is solved numerically by the method of augmented Lagrangeans combined with an operator-splitting technique. Numerical results are given that illustrate the ER-eﬀect for a pure shear mode (Couette ﬂow) and for a more complicated ﬂow structure for an electrorheological clutch. Another mathematical study of the ﬂow of electrorheological ﬂuids and of their constitutive description of Bingham behaviour is given in [35]. First, a boundary value problem for unsteady ﬂow of an electrorheological ﬂuid is formulated. Starting from the most general constitutive function for the stress (8.3.1), the authors assume that t is quadratic in E and aﬃne in d and d/|d| and neglect the term containing E ⊗ E. The resulting constitutive law has the form 1 1 + α32 Ek Ek + α33 Ek Ek dij tij = −pδij + α30 + α31 |d| |d| +α50 (Ei djk Ek + dik Ek Ej ) , (8.3.45) where the coeﬃcients αij are constants that have to fulﬁll α30 ≥ 0 , α31 ≥ 0 , α32 ≥ 0 , α33 ≥ 0 , α32 + 43 α50 ≥ 0 .

(8.3.46)

The restrictions (8.3.46) are deduced from the Clausius–Duhem inequality; they are essential in the further proofs of existence and uniqueness of solutions. Model (8.3.45) is viewed as a combination of Newtonian and Bingham behaviour. As in [67], (8.3.45) does not make sense if |d| = 0, so it cannot be used in the boundary value problem. Instead of this, a variational inequality is formulated for the stress which makes sense. So, the problem is formulated in a variational form. Existence and uniqueness are proved for the solution in the two-dimensional case for any initial data, while in the threedimensional case global existence of a weak solution is proved for small initial data only. In the end of the paper an interesting result is given concerning the estimation of the time when the ﬂuid stops. Both articles emphasize mathematical aspects and do not present identiﬁcation of parameters by experiments. 8.3.2 Constitutive Laws Used in Our (Numerical) Approach In the majority of the experimental evaluations, the constitutive assumptions for ERFs used in the literature are conﬁned to one-dimensional modeling.

312

8 Electrorheological Fluids

The standard conﬁguration is (steady) plane shear ﬂow for which the constitutive equations take the form of a stress-shearing relation [2, 176, 198, 224, 269, 270]). Most popular in the literature are the Bingham, Casson and power law models, depending on the electrorheological material at hand. As mentioned previously, the Casson-like model was introduced in [63]. It generalizes the usual Casson model and includes the Bingham model as a particular case. However, when the more realistic case of ﬁnite electrodes is considered (see Subsect. 8.4.3), the ﬂow is two-dimensional and bidirectional. So, more generally, we have to deal with two-dimensional constitutive equations. Let us recall here the two-dimensional forms of the aforementioned constitutive equations (the indices take the values 1 and 2). • The Bingham model is described by teij = 2η0 dij + 21/2 τy

dij , |te | > 21/2 τy , |d|

(8.3.47)

dij = 0 , |te | ≤ 21/2 τy , (8.3.48) √ where |d| := dmn dnm is one form of the second invariant of d. • The classical Casson model is given by teij = 2η0 dij + 21/2 τy

dij dij + 27/4 (η0 τy )1/2 1/2 , |te | > 21/2 τy , |d| |d|

(8.3.49)

|te | ≤ 21/2 τy .

(8.3.50)

dij = 0 ,

When these equations are used to describe the ERF behaviour, one has to take into account the dependence of the yield stress and, eventually of the viscosity, on the magnitude of the electric ﬁeld. • For the power-law model we have (see [221]) teij = mγ˙ n−1 2dij ,

(8.3.51)

where n > 1 (shear-thickening behaviour). Here γ˙ = 21/2 |d| denotes a generalized shear rate. When n < 1 (shear-thinning or pseudoplastic behaviour) we can no longer use (8.3.51), since for γ˙ → 0 the generalized viscosity mγ˙ n−1 → ∞. The diﬃculty is overcome by modifying the model (8.3.51) through the introduction of a new free parameter γ˙ 0 " mγ˙ 0n−1 2dij , γ˙ ≤ γ˙ 0 , e (8.3.52) tij = mγ˙ n−1 2dij , γ˙ > γ˙ 0 . Here, γ˙ 0 is a constant value of the generalized shear rate below which Newtonian behaviour with viscosity η0 = mγ˙0 n−1 prevails17 . For electrorheological ﬂuids the parameters m, n and γ˙ 0 may depend on the electric ﬁeld. 17

An alternative to regularize 8.3.51 is to replace γ˙ n−1 by (γ˙ n−1 + γ˙0 n−1 ), where γ˙0 is a very small constant.

8.3 Constitutive Laws for the Cauchy Stress Tensor

313

• Let us also generalize (8.3.35)–(8.3.36) and introduce the two-dimensional form of the Casson-like constitutive function as teij = 2η0 dij + β1

2dij 2dij + β2 1/2 , |te | > 2|β1 | , |d| |d|

(8.3.53)

|te | ≤ 2|β1 | ,

(8.3.54)

dij = 0 ,

where the parameters η0 , β1 and β2 are positive and may depend on the magnitude of the electric ﬁeld. The Bingham model and the classical Casson model are obtained from (8.3.53), (8.3.54) if β1 = 2−1/2 τy , β2 = 0 and β1 = 2−1/2 τy , β2 = 23/4 (η0 τy )1/2 , respectively, are chosen. All these models may cause serious mathematical diﬃculties since they are expressed by two-branched functions which are not smooth at the branching point. The presence of the denominator |d| is an obstacle against straightforward numerical modeling of ﬂows in complex geometries because it is diﬃcult to determine a priori where it vanishes. For instance, for models with yield behaviour such as the Bingham, Casson and Casson-like models, it is not possible to determine a priori the yield surfaces (the interfaces which separate a non-deforming solid from a ﬂuid state region) since they have to be determined as part of the solution. Similarly, it is not possible to determine explicitly in what regions of the problem domain γ˙ = γ˙ 0 for the model (8.3.52). Some attempts to overcome this diﬃculty have been proposed. Most of them concern the Bingham model. In 1999 Barnes published a review [21] on models with yield stress. In a special section of the article,“Problems with yield stress and mathematics”, the author mentions important approaches for several complex ﬂow conﬁgurations dealing with yield stress. These are mainly based on the modiﬁcation of the Bingham model in such a way that the mathematical problem concerning the yield stress is avoided. The most important approximations of the Bingham model mentioned by Barnes are the ‘bi-viscosity’ model and the Papanastasiou model. In the bi-viscosity model, the rigid-body character (8.3.48) at low stresses is replaced by a Newtonian ﬂuid behaviour with very high viscosity (ηN η0 ). Then, instead of (8.3.47)–(8.3.48) we have the law teij = 2η0 dij + 21/2 τy

dij , |te | > 21/2 τN , |d|

(8.3.55)

|te | < 21/2 τN ,

(8.3.56)

teij = 2ηN dij , where the constant τN is related with τy by

τy = τN (1 − η0 /ηN ) .

(8.3.57)

The law (8.3.55)–(8.3.56) for ηN → ∞ becomes (8.3.47)–(8.3.48). In Fig. 8.1 one can see how the shear stress depends on the shear rate (one-dimensional

314

8 Electrorheological Fluids

Fig. 8.1. One-dimensional form of the bi-viscosity model (8.3.55)–(8.3.56) (only for positive shear rates)

case) within the bi-viscosity assumption. This model was used in the treatment of squeeze-ﬂow of an electrorheological ﬂuid (see [224]). Papanastasiou introduced in [173] a modiﬁed constitutive equation that smoothes the yield criterion, permitting the numerical treatment of the ﬂow problems based on this model. In our notation his law reads 1 − exp [−n21/2 |d|] (8.3.58) teij = 2η0 + 21/2 τy dij , |d| where the exponent n is a relatively great material parameter that can be determined by experiments. The Bingham law in the unyielded region is recovered from equation (8.3.58) for n → ∞. By contrast to (8.3.47), equation (8.3.58) is not singular since lim teij = 2(η0 + nτy )dij ,

|d|→0

(8.3.59)

and, consequently, it is valid for both the yielded and “unyielded” regions. We should also mention here the so-called alternative Bingham model introduced by Mellgren in [141] 1 e 1/2 (8.3.60) tij = 2η0 + 2 τy dij , (ε/2 + |d|2 )1/2 where ε is a positive material parameter. For ε → 0 (8.3.60) reduces to (8.3.47) but since ε is required to be positive, (8.3.60) is deﬁned for all possible values of dij . As in (8.3.58) the material takes on only a liquid state and therefore does not cause any mathematical diﬃculties.

8.3 Constitutive Laws for the Cauchy Stress Tensor

315

Fig. 8.2. Comparison (one-dimensional form) between the Bingham constitutive function (8.3.27), (8.3.28) and the modiﬁed Bingham models (Papanastasiou’s model, alternative Bingham model) for diﬀerent values of the parameters n and , respectively

In Fig. 8.2 one can see how the models (8.3.58) and (8.3.60) approximate the Bingham model in the one-dimensional case. The yield stress τy functions only as a material parameter without the signiﬁcance of a yield stress. Relations (8.3.58) and (8.3.60) may be seen not only as mathematical approximations of the Bingham model. In [20] a revolutionary but controversial idea in rheology was introduced according to which the concept of yield stress should be seen only as an idealization since “given accurate measurements, no yield stress exists”. So, the viscosity is always ﬁnite. This point of view was supported by some experiments done with rheometers which allow stress measurements for very low shear rates. When the experiment was performed for lower shear rates (for the same material), the value of the yield stress was found to be smaller. Consequently, from this point of view, models (8.3.58) and (8.3.60) are closer to reality. Let us denote by f and g the functions “shear stress vs. shear rate” appearing in the one-dimensional versions of (8.3.58) and (8.3.60),viz., 1 − exp(−n|x|) x, (8.3.61) f (x) = η0 + τy |x| 1 x. (8.3.62) g(x) = η0 + τy (ε + x2 )1/2 If we seek n and ε such that the slopes of the curves are equal in x = 0 i.e. f (0) = g (0) (where f (0) := f (0+ ) = f (0− )) then we obtain n = 1/(ε)1/2 . For n and ε satisfying this relation, the curves for f and g are very close (see Fig. 8.2) showing very similar behaviour.

316

8 Electrorheological Fluids

One can apply the ingenious ideas of the models (8.3.58) and (8.3.60) to avoid the singular behaviour of the stress to the Casson and Casson-like models, too and even to the power-law model. However, when we use these functions to describe the electrorheological ﬂuids, the material parameters η0 , τy and n, ε, respectively, may depend on the electric ﬁeld. For a spacedependent electric ﬁeld the constitutive equations modiﬁed as in (8.3.58) become too complicated due to the presence of the exponential. Due to its simplicity, (8.3.60) is easier to handle than (8.3.58). Let us show how this same regularization can be applied to the power-law model: for instance, in the one-dimensional case of (8.3.52), one may consider instead of the twobranched function (we use the notation of (8.3.17))

˙ > γ˙ 0 , m|γ| ˙ n−1 γ˙ , |γ| (8.3.63) τ= mγ˙ 0n−1 γ˙ , |γ| ˙ ≤ γ˙ 0 , the form τ = m(ε + γ˙ 2 )(n−1)/2 γ˙ ,

(8.3.64)

which eliminates the branching (for illustration and comparison with the original model (8.3.63) see Fig. 8.3). This law was used by Hutter (see [97, 98, 99]) to derive the generalized Glen law in glaciology. As γ˙ → 0 this law exhibits Newtonian behaviour. We found this approach very appropriate for our study. Even though the boundary value problem is quite diﬀerent from ours, recent work of Hild et al. [89] ought to be mentioned who applied the Bingham model to modeling landslides. In their model the viscosity coeﬃcient and the yield stress depend on density which in turn is time- and space-dependent. This makes this approach interesting also for our case. By using variational methods the authors study the blocking property of the ﬂow and describe the rigid zones and the stagnant regions (which are stuck on the boundaries)

Fig. 8.3. Comparison between models (8.3.63) and (8.3.64)

8.4 Applications: Channel Flow of ERFs

317

for certain boundary value problems. However, they explicitly ﬁnd the yield surfaces only for one-dimensional cases. Now synthesizing all the previous issues we will introduce two constitutive models for electrorheological ﬂuids which will further be used in the numerical approach: the alternative Casson-like model (which contains also the alternative forms of the Bingham and Casson models), teij = 2η0 dij + β1 (E)

2dij 2dij + β2 (E) , (δ + |d|2 )1/2 (δ + |d|2 )1/4

(8.3.65)

and the alternative power-law model teij = m(E)(δ + 2|d|2 )(n(E)−1)/2 2dij ,

(8.3.66)

where E = (E12 + E22 )1/2 is the electric ﬁeld modulus and δ is a small positive material parameter. With these two models it is possible to cover a large class of electrorheological materials. They are suitable to numerical simulations and consistent with the phenomenological approaches presented in the previous section. Namely, the alternative Casson-like model may be seen as a particularization of (8.3.39) if we take α20 = α21 = α22 = α30 = α50 = α51 = α52 = 0 ,

(8.3.67)

α31 = β1 , D02 = δ ,

(8.3.68) (8.3.69)

α32 = β2 ,

while the alternative power-law model is included in (8.3.13) as a particular case; it is similar to a particularization of (8.3.9)1 (for α1 = α2 = α5 = 0) with one small deviation: the function m in (8.3.66) depends only on the magnitude of the electric ﬁeld and its expression has to be determined for each ER-material; the dependence of α3 (with the coeﬃcients α31 = α33 = 0) on the magnitude of the electric ﬁeld, takes the particular form α30 + α33 E 2 . For both models the dependence of the coeﬃcients, β1 , β2 for the Cassonlike model and m, n for the power-law model, on the electric ﬁeld are established for a certain ERF with the aid of the experimental data. The measurements are usually performed with rotational viscometers based on the Couette system (made by two concentric cylinders or plate-plate geometry). These devices provide graphs “shear stress vs. shear rate” for diﬀerent values of the electric ﬁelds. The material parameters are obtained from the measured data by ﬁtting techniques, and they are given in tables for diﬀerent values of the electric ﬁeld [2, 63, 269]. By interpolation we can obtain the desired functions.

8.4 Applications: Channel Flow of ERFs under Homogeneous and Inhomogeneous Electric Fields In general, the working behaviour of devices using ER ﬂuids is classiﬁed by three fundamental modes: shear, ﬂow and squeeze [224]. In the ﬂow mode,

318

8 Electrorheological Fluids

which is also called Poiseuille ﬂow (the ﬂow occurs under the eﬀect of an axial pressure), it is assumed that the two electrodes are ﬁxed. The study presented in this section is focused on a special case of the ﬂow mode, the steady pressure-driven ﬂow of electrorheological ﬂuids under isothermal conditions (Θ = const.) in a plane channel under electric ﬁelds produced by several kinds of electrodes. A number of ER equipments, including valves, dampers and actuators is based on this conﬁguration. In the remaining part of this chapter we assume incompressible ﬂuids, i.e., (8.4.1) dkk = vk,k = 0 , where vi := x˙ i . Then the pressure is an unknown of the problem which will be determined as a consequence of the constraint (8.4.1) up to a constant. As mentioned already in Subsect. 8.2.7 we assume (8.2.122) to be valid in the description of ERFs. The ﬁrst ensuing three subsections treat the case of the electrodes ﬂush with the channel walls, parallel to the ﬂow as illustrated in Figs. 8.4–8.7. This case was tackled both theoretically and experimentally. The last subsection is concerned with some more complex conﬁgurations in which the electrodes protrude into the channel (decreasing the channel height) or retreat from the channel (enlarging the channel height). They can still be parallel to the ﬂow (see Fig. 8.53) or oblique (see Fig. 8.54). Another interesting case deals with crenated electrodes (see Figs. 8.55–8.57). Due to the complexity of these last mentioned cases, no theoretical results are yet available for them. We will present the most important experimental results given in the literature and will make some comments about an eventual theoretical modelling and numerical implementation of such cases. 8.4.1 Formulation of the Problem – Electrodes Flush with the Channel Let Ox1 x2 be a Cartesian coordinate system. We consider an inﬁnitely long channel of height 2h made by two inﬁnite parallel planes of zero thickness. These planes are situated at x2 = −h, x2 = h, respectively. Along the channel walls, ﬁnite electrodes, charged with diﬀerent potentials are placed. They may be disposed in various conﬁgurations as one can see in the examples illustrated in Figs. 8.4–8.7. If the electrodes were of inﬁnite length (as in Fig. 8.8), the electric ﬁeld would be homogeneous. However, in every realistic application the electrodes are ﬁnite and the ﬁeld is inhomogeneous, especially close to the edges of the electrodes. The electrodes are isolated outside the channel with a dielectric material having constant electric permittivity ε1 . The ERmedium inside the channel has electric permittivity ε2 which is also a constant according to (8.2.122) and to the fact that the density and temperature are also constant. Since the problem is two-dimensional the indices i and j used henceforth take only the values 1 and 2.

8.4 Applications: Channel Flow of ERFs

319

Fig. 8.4. Conﬁguration with two ﬁnite electrodes of equal lengths (n1 = n2 = 1, V1 = 2V , V2 = 0)

Fig. 8.5. Conﬁguration with two equal but shifted ﬁnite electrodes (n1 = n2 = 1, V1 = 2V , V2 = 0)

Fig. 8.6. Conﬁguration with ﬁnite electrodes interrupted by electrically neutral walls (periodic structure) (n1 = n2 = 3)

Fig. 8.7. Conﬁguration with ﬁnite electrodes interrupted by electrically neutral walls (general case) (n1 = 3, n2 = 2)

Equation (8.2.90) is equivalent with the statement that Ei is the gradient of a scalar function (8.4.2) Ei = −ϕ,i . Here, ϕ is the electric potential. In the further formulation of the boundary value problem (BVP) one has to distinguish between the two cases of conducting and non-conducting ER-ﬂuids. More precisely, the equations from which the electrical unknowns of the problem are determined are diﬀerent in the part of the domain inside the channel: −∞ < x1 < ∞, |x2 | < h. In the domain outside the channel, −∞ < x1 < ∞, |x2 | > h, the formulation is the same in both cases. The boundary conditions are formulated identically for the two cases in the whole domain. 1. Electrically conducting ﬂuids In this case the electrical part of the BVP in the channel domain is given by the divergence of the second Maxwell–Minkowski equation (8.2.91) ∂Dj,j + (Dj,j vk ),k + Jj,j = 0 , ∂t

−∞ < x1 < ∞,

|x2 | < h ,

(8.4.3)

where the electric charge Q was eliminated by using (8.2.4). It is worth mentioning that in this case Q is not a variable that can be given a priori but it has to be determined from (8.2.4) after the calculation of the electric ﬁeld.

320

8 Electrorheological Fluids

From relations (8.2.117), (8.2.118) and (8.2.122), from the incompressibility condition (8.4.1) and from (8.4.2) it follows that ε2

∂ϕ,jj + ε2 ϕ,jjk vk + σ100 ϕ,jj = 0 , ∂t

−∞ < x1 < ∞,

|x2 | < h . (8.4.4)

Since we treat only the stationary case, the ﬁrst term on the left-hand side of (8.4.4) vanishes. As one can see from (8.4.4), due to the presence of the velocity in this equation, the electrical problem is explicitly coupled with the mechanical problem. 1. Electric non-conducting ﬂuids When Jj = 0, then the electric potential is calculated from the third Maxwell-Minkowski equation (8.2.4). Using (8.2.117), (8.2.122) and (8.4.2) one obtains the Poisson equation ϕ,jj = −

Q . ε2

(8.4.5)

In general the electric charge vanishes in electrorheological ﬂuids since only uncharged ﬂuids are treated here18 . Consequently the potential must fulﬁll the Laplace equation in the channel domain ϕ,jj = 0 ,

−∞ < x1 < ∞ ,

|x2 | < h .

(8.4.6)

In the domain outside the channel we assume that the material is electrically non-conducting and that its electric charge can be neglected. Therefore equation (8.4.6) is valid also in the domain −∞ < x1 < ∞, |x2 | > h. We must specify now the boundary conditions. The following Dirichlet boundary conditions are given on the part of the boundary where the electrodes are placed ϕ(x1 , h) = Vi1 , x1 ∈ Iiel1 , ϕ(x1 , −h) = Vi2 , x1 ∈

Iiel2

(8.4.7) ,

(8.4.8)

where i1 = 1, 2, . . . , n1 , i2 = n1 + 1, n1 + 2, . . . , n1 + n2 and n1 , n2 are the numbers of electrodes placed on the upper and lower walls of the channel respectively. Moreover, Iiel1 , Iiel2 are the interval domains of the x1 -coordinate of the electrodes placed on the upper and lower walls of the channel, respectively. Using (8.2.117) and (8.4.2) in (8.2.88), we see that the jump conditions of the electrode-free boundary parts imply the following equations for the normal derivative of ϕ: 18

In our continuum mechanical modeling the assumption of charge neutrality is guaranteed. In a description at the mesoscale or microscale level this would not be fulﬁlled because in ER-ﬂuids free ions can be present.

8.4 Applications: Channel Flow of ERFs

ε2 ϕ,2 (x1 , h− ) = ε1 ϕ,2 (x1 , h+ ) ,

x1 ∈ (−∞, ∞) \

:

Iiel1 ,

321

(8.4.9)

i1

ε2 ϕ,2 (x1 , −h+ ) = ε1 ϕ,2 (x1 , −h− ) ,

x1 ∈ (−∞, ∞) \

:

Iiel2 , (8.4.10)

i2

while from (8.2.86) the following continuity conditions in the tangential derivative are obtained : ϕ,1 (x1 , h+ ) = ϕ,1 (x1 , h− ) , x1 ∈ (−∞, ∞) \ Iiel1 , (8.4.11) i1

−

ϕ,1 (x1 , −h ) = ϕ,1 (x1 , −h ) , +

x1 ∈ (−∞, ∞) \

:

Iiel2 .

(8.4.12)

i2

To specify the jump of the components of the electric ﬁeld across x2 = ±h for x1 ≤ 0, we use the upper index “+” to indicate the limit as x2 tends to ±h from positive values of (x2 ∓ h) and the upper index “−” to indicate the limit as x2 tends to ±h from negative values of (x2 ∓ h). Moreover we may choose (8.4.13) lim ϕ(x1 , x2 ) = 0 , −∞ < x1 < ∞ . x2 →±∞

It is important to remark that for non-conductive ER-ﬂuids (σ1 = 0) we deal with a boundary value problem in which the electric ﬁeld is completely separated from the mechanical ﬁelds19 . Consequently, the solution for the electric ﬁeld can be independently attacked and then used in a second step in the mechanical problem. In the remaining part of the chapter we will study only the case of non-conductive electrorheological ﬂuids. In order to formulate a boundary value problem for the mechanical part we recall the balance of momentum (8.2.63) for a steady ﬂow ρvj vi,j − (tij + Ei Dj ),j − ρFiext = 0 .

(8.4.14)

Using (8.2.107) and (8.2.117) in the last equation, we obtain ρvj vi,j − (−pδij + teij + ε2 Ei Ej ),j − ρFiext = 0 .

(8.4.15)

Using (8.4.2) and (8.4.6), we obtain (Ei Ej ),j = 12 (Ej Ej ),i . 19

(8.4.16)

Of course, the fact that the ﬂow does not aﬀect the electric ﬁeld in this case is equally a consequence of other assumptions such as the neglect of the time derivatives of the strain rate in the constitutive equations (so that (8.2.104) could follow from the entropy inequality which means that the only remaining pure electromagnetic dependent quantity, Dj , cannot depend on the rate of strain tensor dij that is, apart from ρ and Θ the only independent mechanical quantity), the neglect of the electric charge and the consideration of constant density and isothermal conditions.

322

8 Electrorheological Fluids

If we suppose that the external force Fiext is conservative then we may incorporate it into the pressure. Doing this and substituting (8.4.16) in the momentum balance, we obtain − p,i + teij,j + 12 ε2 (Ej Ej ),i = ρvj vi,j .

(8.4.17)

Substituting in (8.4.17) a constitutive function for the Cauchy stress tensor and the solution for the electric ﬁeld, we obtain for plane ﬂow two equations which together with (8.4.1) may be used to determine the three unknowns: the components of the velocity vector and the pressure. Unlike the electrical problem, the domain for the mechanical problem is restricted to the channel. We assume the no-slip and impermeability conditions on the channel walls v1 (x1 , ±h) = 0 ,

v2 (x1 , ±h) = 0 ,

−∞ < x1 < ∞ .

(8.4.18)

8.4.2 Particular Case – Inﬁnitely Long Electrodes If we consider in (8.4.7), (8.4.8) that n1 = n2 = 1, I1el = I2el = (−∞, ∞) and V1 = 2V , V2 = 0 we obtain the conﬁguration sketched in Fig. 8.8. The solution of the electrical problem in this case can be easily derived. It is ϕ(x1 , x2 ) =

V x2 + V , h

−∞ < x1 < ∞ ,

|x2 | ≤ h .

(8.4.19)

Consequently, the electric ﬁeld has the form E1 = 0 ,

E2 = −

V , h

(8.4.20)

everywhere in the channel. The magnitude of the electric ﬁeld is now E = V /h. The dielectric material outside the channel has no inﬂuence on the solution. Another consequence of the fact that the electric ﬁeld is constant

Fig. 8.8. Conﬁguration with inﬁnite electrodes

8.4 Applications: Channel Flow of ERFs

323

is that the ﬂow problem will be uni-directional, i.e. the only non-vanishing velocity component varies in the ﬂow direction, provided, of course, that the inlet and outlet boundary conditions agree with this. We may assume ﬁrst that v1,1 = 0 , v2,1 = 0 . Then, from the continuity equation it follows that v2,2 = 0. If we take into account the impermeability condition on the walls, we obtain that the x2 -component of the velocity vanishes in the whole domain. So, we have v1 = v1 (x2 ) , v2 = 0 .

(8.4.21) (8.4.22)

Consequently, the only non-vanishing component of the rate of strain tensor is D12 = D21 = 12 v1,2 (x2 ) , (8.4.23) and from (8.2.94) and (8.2.95) it follows that all the components of the stress tensor are independent of x1 . By applying all these considerations in the momentum balance (8.4.17) we obtain −p,1 = −te12,2 , −p,2 + te22,2 = 0 .

(8.4.24) (8.4.25)

From the last equation it follows that the quantity −p + te22 can be only a function of x1 . However, as just concluded above, te22,1 = 0 , hence also p,1 is only a function of x1 . Since the right-hand side of (8.4.24) is not a function of x1 , neither is the left-hand side, and, therefore, p,1 = (p − te22 ),1 should be constant. If we denote this constant by k, it follows from (8.4.24) and (8.4.25) that p = kx1 + te22 + c1 , te12,2 = k .

(8.4.26) (8.4.27)

The constant k (k ≤ 0) is the pressure gradient in the x1 -direction. The constant c1 can be determined if a boundary condition for p is given. By assigning the constitutive functions (8.3.65) and (8.3.66) with δ = 0 to teij , one can determine analytically the solution of the problem formulated above. With this choice of the parameter δ, in a viscometric ﬂow, (8.3.65) reduces to the Casson-like model (8.3.35)–(8.3.36)20 introduced by Eckart in [63], while (8.3.66) reduces to the classical power-law model (8.3.33)–(8.3.34). The coeﬃcients β1 , β2 , m and n are constant for a certain value of the electric ﬁeld E, namely of the electric potential V . 20

Unlike in the Casson-like model we choose as a simpliﬁcation β0 = 0. This is equivalent to saying that β0 = 0 is included in the viscosity η0 .

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8 Electrorheological Fluids

The Casson-Like Model This case is solved here following Eckart [63] (see also [64]). First, let us recall the constitutive equations (8.3.35)–(8.3.36) (with the notation according to the previous paragraph: γ˙ is replaced by v1,2 and τ CL is replaced by te12 ) 1/2

te12 = η0 v1,2 + 21/2 β1 + 21/4 β2 v1,2 , te12 > 21/2 β1 ≥ 0 , v1,2 > 0 ,

(8.4.28)

te12 = η0 v1,2 − 21/2 β1 − 21/4 β2 (−v1,2 )1/2 , te12 < −21/2 β1 ≤ 0 , v1,2 < 0 .

(8.4.29)

Integrating (8.4.27) with respect to x2 it is found that te12 = kx2 + C2 ,

|te12 | > 21/2 β1 .

(8.4.30)

Since k < 0, the shear stress te12 is a linear decreasing function of x2 . Assuming that the shear stress at the lower wall τw = −kh + C2 is greater than the yield stress 21/2 β1 , one can distinguish three regions in the channel: • one where the shear stress is greater than or equal to the yield stress: −h ≤ x2 ≤ xi , with xi deﬁning the place where te12 = 21/2 β1 ; the velocity in this region is increasing with respect to x2 , hence the upper index “i ” is used; • one where the material becomes solid and the velocity denoted by v p is constant: xi ≤ x2 ≤ xd , with xd deﬁning the place where te12 = −21/2 β1 ; this region is called the plug zone, so the upper index “p” is used; • one where the shear stress is less than or equal to −21/2 β1 : xd ≤ x2 ≤ h; here the velocity is decreasing and the upper index “d ” is used. The places xi and xd can readily be found from (8.4.30) as xi2 = −

21/2 β1 C2 + , k k

xd2 = −

21/2 β1 C2 − . k k

(8.4.31)

The symmetry with respect to the axis x2 = 0 of the boundary value problem implies xi = −xd . Hence, from (8.4.31) it follows that C2 = 0 .

(8.4.32)

By inserting (8.4.28) into (8.4.30) and by taking into account (8.4.32) one arrives at the diﬀerential equation for v1i i v1,2 +

21/4 β2 i 1/2 −kx2 + 21/2 β1 (v1,2 ) + =0. η0 η0

(8.4.33)

i Employing the substitution v1,2 = (ui )2 in the previous equation, solving the quadratic equation for ui and using the inverting formula v1i = , iemerging 2 (u ) dx2 + C3i , one derives the general solution for v1i in the form

8.4 Applications: Channel Flow of ERFs

325

k 2 2−1/2 β22 − 21/2 η0 β1 x + x2 2η0 2 η02 3/2 4 2−3/4 β2 2−3/2 β22 kx2 − 21/2 β1 ∓ + + C3i . (8.4.34) 3 k η02 η0

v1i (x2 ) =

One proceeds analogously for negative shear rates: insertion of (8.4.29) and d = −(ud )2 and then of the (8.4.32) into (8.4.30), use of the substitution v1,2 , d 2 d d inverting formula v1 = − (u ) dx2 + C3 yields k 2 2−1/2 β22 − 21/2 η0 β1 x − x2 2η0 2 η02 3/2 4 2−3/4 β2 2−3/2 β22 kx2 + 21/2 β1 ∓ − + C3d . (8.4.35) 3 k η02 η0

v1d (x2 ) =

i/d

To get rid of the constants C3 , the boundary condition (8.4.18) are imposed on (8.4.34), (8.4.35), so k 2−1/2 β22 − 21/2 η0 β1 (x22 − h2 ) + (x2 + h) 2η0 η02 3/2 4 2−3/4 β2 2−3/2 β22 kx2 − 21/2 β1 ∓ + 3 k η02 η0 3/2 4 2−3/4 β2 2−3/2 β22 kh + 21/2 β1 ± − , 3 k η02 η0

v1i (x2 ) =

k 2−1/2 β22 − 21/2 η0 β1 (x22 − h2 ) + (h − x2 ) (2η0 ) η02 3/2 4 2−3/4 β2 2−3/2 β22 kx2 + 21/2 β1 ∓ − 3 k η02 β0 3/2 4 2−3/4 β2 2−3/2 β22 kh + 21/2 β1 ± − . 3 k η02 η0

(8.4.36)

v1d (x2 ) =

(8.4.37)

To deduce the correct signs in (8.4.36), (8.4.37) two additional conditions are needed. Notice that the velocity function should be continuously diﬀerentiable p = 0 in the plug zone. This leads to for −h < x2 < h and that v1,2 !

!

i d v1,2 (x2 = xi2 ) = 0 , v1,2 (x2 = xd2 ) = 0 .

(8.4.38)

Using (8.4.38) one ﬁnally determines the following solutions for the velocities

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8 Electrorheological Fluids

k 2−1/2 β22 − 21/2 η0 β1 (x22 − h2 ) + (x2 + h) 2η0 η02 3/2 4 2−3/4 β2 2−3/2 β22 kx2 − 21/2 β1 − + 3 k η02 η0 3/2 −3/2 2 −3/4 β2 2 β2 42 kh + 21/2 β1 + − , 3 k η02 η0

v1i (x2 ) =

21/2 β1 , (8.4.39) k 3/2 2 k 1 2−3/4 β2 21/2 β22 21/2 β1 =− − h+ 2η0 k 6 k η02 3/2 4 2−3/4 β2 2−3/2 β22 k h + 21/2 β1 + − 3 k η02 η0 1/2 −1/2 2 1/2 β2 2 21/2 β1 2 β1 2 β1 ≤ x , (8.4.40) + ≤ − h + , 2 η02 k k k −h ≤ x2 ≤

v1p

k 2−1/2 β22 − 21/2 η0 β1 (x22 − h2 ) + (h − x2 ) (2η0 ) η02 3/2 4 2−3/4 β2 2−3/2 β22 kx2 + 21/2 β1 − − 3 k η02 β0 3/2 −3/2 2 −3/4 β2 2 β2 42 kh + 21/2 β1 + − , 3 k η02 η0

v1d (x2 ) =

−

21/2 β1 ≤ x2 ≤ h . k

(8.4.41)

Since the Casson-like model includes√the Newtonian (β1 = β2 = 0), Bingham (β2 = 0) and Casson (β2 = 2 η0 β1 ) behaviours, the given solution may be particularized for all these types of ﬂuids. Having found the analytical solution for the velocity ﬁeld one can calculate the volumetric ﬂow rate b h Q=

h v1 (x2 )dx2 dx3 = b

0 −h

v1 (x2 )dx2 ,

(8.4.42)

−h

where b is the width of the channel. The following formula relates the pressure gradient k = p,1 with the volumetric ﬂow rate Q:

8.4 Applications: Channel Flow of ERFs

327

√ √ 3 3 √ 6 √ 2 2 3 b 4 2 η β β + 80 2 η β − 4 2 β − 120 2 k h η0 β1 40 0 1 2 0 1 2 4 120 η0 k 2 √ √ −120 2 η02 β12 β22 + 60 2 k 2 h2 η02 β22 + 80 k 3 h3 η03 + 211/4 kh Z1 η0 β23 −48 · 21/4 k 2 h2 Z1 η02 β2 − 219/4 kh Z1 η02 β1 β2 + 29/4 Z1 β25 (8.4.43) −221/4 Z1 η0 β1 β23 + 225/4 Z1 η02 β12 β2 ,

Q=

√ √ where Z1 := 2 β22 −4 khη0 −4 2 η0 β1 . Equation (8.4.43) yields, of course, the volumetric ﬂow rate of a Newtonian ﬂuid, a Bingham ﬂuid and a Casson ﬂuid if the corresponding particular values of the coeﬃcients βi , i = 1, 2 are selected. If we use a procedure in which Q is the input value and we want to calculate the pressure drop as output, we have to invert formula (8.4.43) in order to obtain the pressure gradient for non-vanishing values of the electric ﬁelds. Analytically, this is not possible, but using the software Mathematica [268] we may obtain k as a function of the volumetric ﬂow rate Q and of the uniform electric ﬁeld E. The pressure gradient for a vanishing electric ﬁeld (when β1 = β2 = 0) is 3Qη0 , 2bh3

(8.4.44)

Q = 43 hbv0 ,

(8.4.45)

k=− and

where v0 = v2 (0) is the maximum velocity in the channel. The above formulas may be used to evaluate approximately the pressure drop in conﬁgurations with long ﬁnite electrodes when neglecting end eﬀects. Namely, on the electrode-free part of the channel, the pressure drop is calculated by applying the formula for a vanishing electric ﬁeld (8.4.44) while on the part where the electrodes are placed, the pressure drop is calculated by applying the inversion of formula (8.4.43) for a constant electric ﬁeld, where the value of the electric ﬁeld is chosen to be the value established in the homogeneous region, in the middle of the electrodes, far from the electrode edges. Similar methods were used by Eckart in [64] and by Wunderlich in [270] in order to compare the analytical results with the experimental results. The relatively small deviations of the analytical plots from the measured values are due to the neglect of the electric ﬁeld in the region outside the electrodes and of the inhomogeneity of the electric ﬁeld in the vicinity of the electrode edges. The Power-Law Model If we solve problem (8.4.21)–(8.4.27) for a power-law ﬂuid (8.3.33)–(8.3.34) we obtain (on assuming k < 0) the following velocity ﬁeld

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8 Electrorheological Fluids

v1 (x2 ) = where k1 = (−k/m)

1/n

k1 n (n+1)/n (h − |x2 |(n+1)/n ) , n+1

(8.4.46)

. Now, calculating the volumetric ﬂow rate yields Q=

2(n + 1) v0 hb , 2n + 1

(8.4.47)

where

k1 n (n+1)/n h (8.4.48) n+1 is the maximum velocity in the channel. Inverting this formula, we obtain n Q(2 + 1/n) m . (8.4.49) k=− hb h v0 = v1 (0) =

The model presented in this subsection was so far used to describe the ﬂow of an ERF in a channel (see [63]). When interpreting the experimental results in slit ﬂows, it was routinely assumed that the electric ﬁeld is constant and that it determines a shear ﬂow in the channel [4, 5, 198, 207, 270]. In other words, the model with inﬁnite electrodes was used as an approximation for the real case with ﬁnite electrodes. In our view this is not a realistic approximation in all cases since it neglects the electrode end eﬀects which may considerably aﬀect the ﬂow as we will show in the next subsection. 8.4.3 Electrodes of Finite Length As we mentioned in the Introduction, an impediment to overcome to make an industrial exploitation of the ER-eﬀect on a large scale possible is the very high voltage requirements necessary to obtain the desired increase in viscosity. There are attempts to increase the electrorheological eﬀect by modifying either the surface or the shape and position of the electrodes relative to the ﬂow geometry in such a way that inhomogeneities in the electric ﬁeld are introduced [3, 4, 5, 6, 7, 33, 81, 108, 150, 166, 269, 270]. All these experimental investigations demonstrated that application of non-uniform electric ﬁelds may lead to more eﬃcient eﬀects on the ﬂow (than with homogeneous electric ﬁelds). The necessity of models which reproduce the ER behaviour for inhomogeneous electric ﬁelds was formulated in certain ﬁelds of applications [36, 114], the purpose being an accurate description of the experiments performed in order to improve the performance of ER devices. Nevertheless, in most theoretical approaches of ERF ﬂows the electric ﬁeld is only a constant parameter. In channel ﬂow this is an analytical consequence of the fact that the electrodes are considered to be inﬁnite while in cylindrical Couette-type conﬁgurations this is an assumption that can be made when the ﬂuid channel is small compared with the radius of the inner cylinder. Exceptions are the theoretical

8.4 Applications: Channel Flow of ERFs

329

results for a radial conﬁguration obtained by Atkin et al. [16, 17] where the electric ﬁeld is slightly inhomogeneous in the radial direction. Rajagopal ˇka in [196] and Eckart in [63] developed a theoretical framework and R˚ uˇ zi c which allows for variable ﬁeld strength. However, in such a case the electric ﬁeld has to be determined from the Maxwell equations. The aim of the study presented in this subsection is twofold. On the one hand, we consider it important to give a more realistic modeling by taking into account the inhomogeneity eﬀects which appear in the vicinity of electrode edges. Comparing this with the case of plane shear ﬂow exposed to inﬁnite electrodes, where the electric ﬁeld is simply a constant, the nonuniform electric ﬁeld will cause here inhomogeneities in the ﬂow too which, as a consequence, will be non-viscometric. Consequently, both components of the velocity are variables of the problem and they depend on both coordinates. On the other hand, our intention is to examine numerically how the ER-eﬀect can be enhanced by a space-dependent electric ﬁeld. Let us return to the problem formulated in Subsect. 8.4.1 for electrically non-conducting ﬂuids. It consists of the electrical problem which may be solved independently and the mechanical problem, equations which contain terms based on the electric ﬁeld components, namely on the solution of the electrical problem. As one can see, the electric ﬁeld is a key element since its dependence on the space coordinates determines the kinematical character of the ﬂow. In order to gain a better understanding of the end eﬀects of the electrodes on the electric ﬁeld inhomogeneity in the channel and then on the ﬂow, one needs to investigate ﬁrst the simplest conﬁguration of electrodes. In [246] and [247] the distribution of the electric potential around two long electrodes charged with diﬀerent potentials in a symmetric, an anti-symmetric and a non-symmetric way was investigated. The term “long electrode” denotes in these works either a semi-inﬁnite electrode or a ﬁnite electrode of a certain length chosen such that the two far edges of the electrode do not interact. The solutions were found (semi)-analytically and they were constructed with the use of the Wiener–Hopf (WH) technique21 . With the help of the WH-method it was possible to set the singularities at the tips of the electrodes explicitly in evidence, which in a numerical solution must be approximately accounted for in rather costly mesh reﬁnements. Furthermore, the obtained analytical solution was used to test and validate the numerical solution ob21

We mention that the problem is solved in a bounded domain in the x2 -direction, where the upper and lower boundaries consist of two inﬁnite grounded electrodes situated at a distance H > h from the x1 -axis. If H is suﬃciently large, this is equivalent with the usual inﬁnity conditions which means vanishing potential at x2 = ±∞. For smaller H, this conﬁguration can still be easily realized in practice. We mention that the two grounding electrodes at x2 = ±H are needed for technical reasons when solving the WH-problem, for otherwise, i.e., when no grounding electrodes are present no solution could be found (see [247] for a detailed explanation). Physically, this is no restriction because the channel will always be earthed and the system can always be looked at for large H.

330

8 Electrorheological Fluids

tained using the software Femlab [47] (which is a powerful tool for solving partial diﬀerential equations by applying the ﬁnite element method). Apart from the purely analytical and numerical approach to determine the solution of the electrical problem for the conﬁgurations with two long electrodes, it may be more advantageous to use a mixed approach, in which one uses the analytical solution in the numerical program and so reduces the domain of numerical approach. Concretely, we can impose in the numerical program Dirichlet conditions for the electric problem on the whole channel walls deduced from the analytical solution so that the domain of the electric problem is equally reduced to the channel as for the mechanical problem. Since the mathematical expressions of the analytical solutions for the electric potential on the walls are too complicated and create diﬃculties when trying to introduce them directly into Femlab, we will interpolate them and use polynomials which accurately approximate the exact solution. In this way space memory and computing time can be saved. We call this approach numerical analytic. When we treat conﬁgurations with two short electrodes or with more than two electrodes then the electric ﬁeld will be calculated numerically. This is the so-called completely numerical approach. Let us recall the equations of the mechanical problem: −p,i + teij,j +

1 2

ε2 (Ej Ej ),i = ρvj vi,j ,

vi,i = 0 ,

(8.4.50) (8.4.51)

with the boundary conditions v1,1 (−L, x2 ) = 0, v2 (−L, x2 ) = 0 , p(L, x2 ) = 0 ,

|x2 | ≤ h , |x2 | ≤ h ,

(8.4.52) (8.4.53)

vi (x1 , ±h) = 0 ,

|x1 | ≤ L .

(8.4.54)

The x1 -positions −L and L mark the entrance and the exit of the ﬂuid in the channel, respectively. We assume that L is suﬃciently large; speciﬁcally the inlet boundary is far enough from the electrode edge so that the electric ﬁeld is negligible. Consequently, one may impose there the velocity as in a uni-directional channel ﬂow (described in Subsect. 8.4.2). Beyond the channel exit we assume vanishing pressure. For the extra stress teij we will use models (8.3.65) and (8.3.66) and thus may deﬁne a generalized viscosity ηgen so that teij = 2ηgen Dij . Explicitly, ηgen = η0 +

β1 (E) β2 (E) + 2 1/2 (δ + |D| ) (δ + |D|2 )1/4

(8.4.55)

for the alternative Casson-like model and ηgen = m(E)(δ + 2|D|2 )(n(E)−1)/2 for the alternative power-law model.

(8.4.56)

8.4 Applications: Channel Flow of ERFs

331

The Dimensionless Problem for the Alternative Casson-Like Model ˜i and p˜ according to We introduce the non-dimensional quantities x ˜i , v˜i , E ˜i , p = p0 p˜ , xi = h˜ xi , vi = v0 v˜i , Ei = E0 E (8.4.57) where h, v0 , E0 and p0 are characteristic quantities of the problem: h is half the channel height, v0 is the maximum inlet velocity, E0 is a typical value for the electric ﬁeld and p0 = η0 v0 /h. Application of these transformations to (8.4.50), multiplying the resulting expression with h/p0 and dropping the “tilde”, yields 1 (Ej Ej ),i = Re vj vi,j , (8.4.58) − p,i + 2 (ηgen dij ),j + 2Ma where √ 1 h β2 (E0 E) 1 h β1 (E0 E) + (8.4.59) ηgen = 1 + √ 2 1/2 v0 η0 v0 η0 (nδ + |d| ) (nδ + |d|2 )1/4 is the dimensionless generalized viscosity, Re = ρhv0 /η0 ,

(8.4.60)

Ma = η0 v0 /(hε2 E02 ) ,

(8.4.61)

is the Reynolds number, 2

/v02 .

is the Mason number and nδ = δh The Reynolds number is usually interpreted as the ratio of the inertial force to the viscous force while the Mason number can be interpreted as the ratio of the viscous force to the electrostatic force. We have chosen a non-dimensionalization appropriate to creeping ﬂows for which the Reynolds number takes small values. The dimensionless form of equation (8.4.51), after dropping the “tilde”, remains unchanged. Usually, β1 , β2 vanish when the electric ﬁeld is zero, so that in this case the ﬂuid response is Newtonian. Consequently, at the channel entrance, where the electric ﬁeld is approximately zero, we have before non-dimensionalization v1 (−L, x2 ) = v0 (1 − (x2 /h)2 ) .

(8.4.62)

The non-dimensionalized boundary conditions (8.4.52)–(8.4.54) and (8.4.62) become v1 (−L/h, x2 ) = 1 − x22 , v2 (−L/h, x2 ) = 0 , |x2 | ≤ 1 , p(L/h, x2 ) = 0 , |x2 | ≤ 1 , vi (x1 , ±1) = 0 ,

(8.4.63) (8.4.64)

|x1 | ≤ L/h . (8.4.65)

The BVP is formulated now by equations (8.4.51), (8.4.58), (8.4.59) together with the boundary conditions (8.4.63)–(8.4.65), where Ej is the nondimensionalised solution of (8.4.2)–(8.4.12). It will be solved numerically for the domain |x1 | ≤ L/h, |x2 | ≤ 1 using the commercial software Femlab [47].

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8 Electrorheological Fluids

The Dimensionless Problem for the Alternative Power-Law Model We apply the same procedure as in the previous case, namely we introduce the non-dimensional quantities as in (8.4.57), but now consider p0 = m0 (v0 /h)n0 , where m0 = m(0), n0 = n(0) (which are usually non-zero). The nondimensional momentum balance for a power-law ﬂuid has the same form as (8.4.58) with the Mason number, the Reynolds number and the nondimensional generalized viscosity given by Ma = ηgen =

m0 (v0 /h)n0 , ε2 E02

Re =

ρv02 , m0 (v0 /h)n0

m(E0 E)(v0 /h)n(E0 E) (nδ + 2|d|2 )(n(E0 E)−1)/2 . m0 (v0 /h)n0

(8.4.66)

(8.4.67)

We remark that, since the dimension of m is Pasn and since n, which is dimensionless, is space dependent, the quantity m has variable dimension in space. Even though we can ﬁt the values of m (from the experimental data) and we may obtain the function m(E0 E), this quantity is meaningless from a physical point of view. In order to avoid this problem, we will directly ﬁt the dimensionless function (see (8.4.67)) f (E0 E, v0 /h) = m(E0 E)(v0 /h)n(E0 E)

(8.4.68)

to the data. An example of this operation will be presented later in this subsection. In order to establish the non-dimensional form of the boundary conditions let us apply formula (8.4.46) for vanishing electric ﬁeld which gives the dimensional velocity at the entrance of the channel (n0 +1)/n0 |x2 | . (8.4.69) v1 (−L, x2 ) = v0 1 − h Consequently, the non-dimensional inlet boundary condition has the form v1 (−L/h, x2 ) = 1 − |x2 |(n0 +1)/n0 , v2 (−L/h, x2 ) = 0 .

(8.4.70)

The non-dimensional boundary conditions at the channel exit and on the channel walls are identical with their correspondents from the Casson-like case: (8.4.64) and (8.4.65). The problem formulated by equations (8.4.51), (8.4.58), (8.4.67) together with the boundary conditions (8.4.70), (8.4.64) and (8.4.65), where Ej is the non-dimensionalised solution of (8.4.2)–(8.4.12) will be solved numerically for the domain |x1 | ≤ L/h, |x2 | ≤ 1 using the commercial software Femlab [47].

8.4 Applications: Channel Flow of ERFs

333

a) Numerical Results – the Alternative Casson-Like Model Material and Conﬁguration Properties Used for the Simulations An ER-ﬂuid chosen for simulations is Rheobay TP AI 3565 which is produced by the Bayer Company (Germany). It is water-free and consists of polyurethan particles in silicone oil, some additional additives and an emulgator [2, 270, 23]. In [63], the Casson-like model (8.3.35)–(8.3.36) is recommended for this ﬂuid since this model reproduces well the data measured in a rotational viscometer. The ﬂuid shows Newtonian behaviour in the absence of an electric ﬁeld i.e. β1 = 0 and β2 = 0 for E0 E = 0, (E = (E12 + E22 )1/2 is the modulus of the dimensionless electric ﬁeld). The value η0 = 0.037 Pa·s for the viscosity and the values of the parameters β1 and β2 given in Table 8.1 were obtained in [64] using the experimental plots for creeping ﬂow “stress vs. shear rate”. Table 8.1. Values of the parameters β1 and β2 of (8.3.35) for diﬀerent electric ﬁeld strengths kV ] E0 E [ mm

0.5

1

2

3

β1 [Pa]

19.3

100.32

312.94

582.81

0.44

5.84

9.7

32.21

β2 [Pa·s

1/2

]

Fitting these data we obtained the dependence of the parameters on the dimensional electric ﬁeld E0 E: β1 (E0 E) = α11 (E0 E) + α12 (E0 E)2 , β2 (E0 E) = α21 (E0 E) + α22 (E0 E)2 + α23 (E0 E)3 ,

(8.4.71) (8.4.72)

where the coeﬃcients αij are given in Table 8.2. The functions β1 and β2 are written as series expansions by retaining only the terms up to the second and the third order, respectively. The experimental data are provided for a limited range of the electric ﬁeld and do not Table 8.2. Values of the coeﬃcients α1i and α2i from (8.4.71), (8.4.72) i

1

2

α1i

55.322 Pa·mm/kV

46.946 Pa·(mm/kV)2

α2i

1/2

7.01 Pa·s

·mm/kV

−5.22 Pa·s

3 –

·(mm/kV)

1/2

2

2.15 Pa·s

·(mm/kV)3

1/2

334

8 Electrorheological Fluids

Fig. 8.9. Fitting curves for the dependence of the coeﬃcients β1 and β2 on the electric ﬁeld. The insets show how the ﬁt is extrapolated

cover the range of the electric ﬁeld used in the simulations. We use formulas (8.4.71) and (8.4.72) also to extrapolate the given data for β1 and β2 for larger values of E. Figure 8.9 illustrates how the ﬁtting curves approximate the data and how they are extrapolated. In the subsequent analysis we shall use the alternative bidimensional Casson-like model (8.3.65). Consequently, the generalized viscosity is expressed by a low-degree polynomial function in E. Substituting (8.4.71) and (8.4.72) in (8.4.59), we obtain ηgen = 1 + n11

E E2 E E2 E3 + n12 + n21 1/2 + n22 1/2 + n23 1/2 , dδ dδ d d d δ

δ

(8.4.73)

δ

where we used the short-hand notation dδ = (nδ + dmn dmn )1/2 .

(8.4.74)

The non-dimensional coeﬃcients nij , i = 1, 2, j = 1, 2, 3 are deﬁned as hα11 E0 , v η √0 0 hα21 E0 = √ , v0 η0

n11 = n21

hα12 E02 , v η √0 0 2 hα22 E0 = √ , v0 η0

n12 = n22

√ n23 =

hα23 E03 . √ v0 η0

(8.4.75)

For all graphs that subsequently will be shown we used a geometry characterized by the value H/h = 10, where the x2 -coordinates ±H mark the upper and lower boundaries for the domain of the electrical problem. The value of L/h is always chosen so that the inlet boundary is suﬃciently far from the electrode edge in order to insure a negligible electric ﬁeld at the entrance of the channel. The electric permittivity of Rheobay is ε2 = 1.4 × 10−9 A·s/(V·m) and we choose ε1 = 0.02 ε2 . The density of Rheobay is ρ = 1041 kg/m3 . All simulations except those done to make the comparison with the experimental data from Fig. 8.29 are performed for

8.4 Applications: Channel Flow of ERFs

335

Fig. 8.10. The mesh around the electrode edges. The arrows show the position of the electrode edges

h = 1 mm, E0 = 1 kV/mm and v0 = 0.3 m/s. So, Re = 7.849, , Ma = 0.007, n11 = 5.417, n12 = 4.597, n21 = 11.433, n22 = −8.515 and n23 = 3.506. We considered here nδ = 0.002. We will study later the inﬂuence of this parameter on the solution. The Flow Near the Electrode Ends (long electrodes) Let us ﬁrst illustrate the eﬀect of the electric ﬁeld inhomogeneity produced by the electrode edges on the ﬂow. To do this we solved the problem (8.4.51), (8.4.58), (8.4.73) together with the boundary conditions (8.4.63)–(8.4.65), where Ej is the non-dimensionalised solution of (8.4.2)–(8.4.12) with two long electrodes charged anti-symmetrically, using the numerical analytic approach. We took V = 2 kV. A triangular mesh consisting of approximately 10000 elements (9984) was used. The default mesh is twice reﬁned (with a regular reﬁnement) so that the obtained mesh is characterized by 8 elements between the channel walls in the regions far from the electrode ends. The maximum element size near the vertices given by the electrode ends is taken to be 0.1. A detail of the mesh in the region close to the electrode edges is plotted in Fig. 8.10. We choose linear Lagrange elements for the electric problem and p2-p1 Lagrange elements22 for the ﬂuid problem. We present and describe here the most relevant ﬁelds in an area close to the electrode ends chosen in order to set in evidence the eﬀects ahead, near and after the electrode ends. This is a crossing zone in which the quantities of the studied problem are passing 22

One has quadratic element for the variables vi , i = 1, 2 and linear element for the variable p

336

8 Electrorheological Fluids

from the regime without an electric ﬁeld to a regime with uniform electric ﬁeld through a transition zone with strong inhomogeneities. The ranges for the color bars corresponding to Figs. 8.11, 8.19 and 8.20 were chosen so that the transition zone is illustrated in an especially relevant fashion. Consequently, the values from the zones with the darkest red could be greater than the maximum values indicated on the scale. We mention that the maximum values in a given area surrounding the electrode edges are much larger than the maximum value within a similar area with uniform electric ﬁeld (inside the electrodes, far from the edges). The closest area to the edges is diﬃcult to be described exactly for three reasons. First, the electric ﬁeld is singular there (see [246]). Second, the material properties (β1 and β2 ) are not provided for high values of the electric ﬁeld (see Table 8.1). Third, in practice, the electrodes can not be inﬁnitely thin as approximated here, so E is not singular at the edges but has a large value. Let us show ﬁrst in Fig. 8.11 the electric ﬁeld produced by this conﬁguration. The inhomogeneities of the electric ﬁeld extend over a length of order h. In the vicinity of an order smaller than h/10 around the electrode ends, the electric ﬁeld is very large, its value is exceeding the ranges for which experimental data are provided. In this vicinity, due to the singularity of the electric ﬁeld, the numerical simulation of the electric ﬁeld is not stable (since mesh dependence was observed). However, this vicinity is small in comparison with the characteristic length of the experimental conﬁguration. In Fig. 8.12 we plot the non-dimensional modulus of the velocity, v(x1 , x2 ) = (v12 (x1 , x2 ) + v22 (x1 , x2 ))1/2 , in the vicinity of the electrode edges

Fig. 8.11. Surface plot of the dimensionless modulus of the electric ﬁeld

8.4 Applications: Channel Flow of ERFs

337

Fig. 8.12. Surface plot of the non-dimensional velocity modulus in the vicinity of the electrodes edges (x1 = −20)

Fig. 8.13. Vector plot of the non-dimensional velocity

(x1 = −20). Note the two small domains formed around the electrode ends where the ﬂuid is almost solidiﬁed and the diﬀerence outside and between the electrodes. To have a better view over the ﬂow in the inhomogeneous area, we present in Fig. 8.13 an arrow plot of the velocity ﬁeld in a similar, somewhat smaller area. Immediately before and immediately after the electrode ends, v2 is not negligible; indeed, the transverse velocity inﬂuences the proﬁle of the x1 -component as one can see in Figs. 8.14 and 8.15. It is worth mentioning that the two extreme values of v2 at x1 = −20.2 (ﬁrst positive and then negative) and x = −19.5 (ﬁrst negative and then positive) indicate that the ﬂuid avoids the electrode tips and ﬂows around them. For x1 outside this regime, far ahead of the electrode ends where the proﬁle of v1 is Newtonian the x2 -component of the velocity almost vanishes (see Figs. 8.16 (left) and 8.17). Between the electrodes, far downstream from the electrode edges the ﬂuid velocity assumes the known one-dimensional Casson-like proﬁle (see Figs. 8.16 (right) and 8.17). In Fig. 8.18 we plot the pressure along a part of the channel including the transition area. The distinguished three regions are characterized by diﬀerent pressure gradients. First one can see a region with a very small gradient

338

8 Electrorheological Fluids

Fig. 8.14. Proﬁles of v1 at x1 = −20.2, −20, −19.5, respectively

Fig. 8.15. Proﬁles of v2 at x1 = −20.2, −20, −19.5, respectively

Fig. 8.16. Proﬁles of v1 at x1 = −23.5, −16.5, respectively

Fig. 8.17. Proﬁles of v2 at x1 = −23.5, −16.5, respectively

8.4 Applications: Channel Flow of ERFs

339

Fig. 8.18. Dimensionless pressure in the transition zone (x1 = −20 marks the electrode ends)

corresponding to the Newtonian ﬂuid; then a steep region with a high gradient corresponding to the entrance between the electrodes is followed by the third region with the gradient corresponding to the Casson-like ﬂuid. For x1 < −20.3 and for x1 > −19.5 the gradients may be calculated analytically from the formulas corresponding to the Newtonian ﬂuid and the Casson-like ﬂuid, respectively (see Subsect. 8.4.2). The increase in pressure drop produced within the second, so-called transition zone is important and shows that the inhomogeneity produced by the end eﬀects may be used to obtain an enhancement of the ER-eﬀect. This strong inhomogeneity eﬀect can also be seen by the diﬀerence of the pressure curves for x2 = 0 and x2 = 0.7, respectively. In Fig. 8.19 the second invariant of the rate of strain tensor dij is plotted. As expected, the domains with high values of |d|2 are near the walls and in the electrode region, since the velocity is zero on the walls and the velocity gradients are larger there than at electrode-free boudaries, see Fig. 8.13. Exceptions arise in the vicinity of the electrode ends. There one can see zones with reduced shear rate, which suggests the presence of near solid zones. This behaviour can be attributed to the high electric ﬁeld generated by the electrode edge. In the middle of the channel, |d|2 maintains a small value. In Fig. 8.20 we display the generalized viscosity. As one can see from (8.4.59) and from Fig. 8.9, ηgen is decreasing when |d| is increasing, and it is increasing when E is increasing (since β1 and β2 are increasing with E). The electrode edges can again be identiﬁed; close to them, both E and |d|

340

8 Electrorheological Fluids

Fig. 8.19. Surface plot of dimensionless |d|2

Fig. 8.20. Surface plot of the generalized dimensionless viscosity

give increasing contributions. An interesting shape of the viscosity ηgen ought to be noticed: right between the electrode ends a high viscosity “island” is formed after which it quickly decreases; then it increases again and assumed the known shape of the Casson-like ﬂuid far downstream (with the unyielded region in the middle). Inﬂuence of the Parameters c and δ Figure 8.21 shows the behaviour of the electric ﬁeld outside the electrodes for diﬀerent values of the electric permittivity ratio (c = ε1 /ε2 ). As shown in [246], the electric ﬁeld is stronger in the area without electrodes when the permittivity of the material outside the electrodes is larger than the permittivity of the ER-ﬂuid. The plot also shows that in the small area close

8.4 Applications: Channel Flow of ERFs

341

Fig. 8.21. Line plots of E(x, 0) (continuous line) and E(x, 0.7) (dashed line) for diﬀerent values of c (electrodes end at x1 = −20)

to the electrode edges this eﬀect is reversed. As a consequence, we expect a contribution to the pressure drop from the region outside the electrodes when c is greater than 1. We plot in Fig. 8.22 the pressure in the region where the ERF exits from the electrodes but including also the end of the channel where the pressure takes the reference value p = 0. One can see that the pressure drop immediately at the electrode ends is slightly decreasing with the value of c. This eﬀect may be important in conﬁgurations with short electrodes. For long electrodes the diﬀerence is not relevant because the main contribution to the pressure drop comes from the E = const. region between the electrodes, see Fig. 8.18. In Fig. 8.23 (left) we plot the pressure drops between x1 = 19 or x1 = 21 and the end of the channel, x1 = 35 (where the outlet boundary condition imposes p = 0) versus the electric permittivity ratio (c). These pressures, denoted by p19 and p21 , respectively, were calculated in the middle of the channel height x2 = 0 and at x2 = 0.7. One can see from the ﬁgure that the pressure in the region outside the electrodes does not depend on x2 while a small dependence of the pressure drop on x2 can be observed across the electrode edges. For c greater than 50 the change in pressure drop is negligible. Since the permittivity of the ER-ﬂuid is quite small, in practice c 1. To see the eﬀects for small values of the ratio c we repeat the same plot in Fig. 8.23 (right) but use now for c a logarithmic scale. Another key parameter of the model is δ. This can be viewed as a parameter which ought to be chosen as small as possible in order to recover from equation (8.3.65) the Casson-like model, (equation (8.3.53)). On the other hand, δ can be viewed as a constitutive parameter which can be obtained from equation (8.3.65) by ﬁtting the experimental data.

342

8 Electrorheological Fluids

Fig. 8.22. Pressure (at x2 = 0) over the electrode ends for diﬀerent c

Fig. 8.23. Pressure drop p19 and p21 vs. ratio of the permittivities c on a linear scale (left) and on a semilogarithmic scale (right)

In what follows we study the inﬂuence of nδ = δh2 /v02 (the dimensionless form of δ) on the solution. In Figs. 8.24 and 8.25 we plotted ηgen (|d|2 , E = 2) for diﬀerent values of nδ according to formula (8.4.55). The two ﬁgures diﬀer in the range of the argument |d|2 . One can see from Fig. 8.24 that the choice of nδ (nδ = 0.002) implies that the generalized viscosity diﬀers from the classical generalized Casson-like viscosity (for which nδ = 0) only for |d|2 < 0.01. Figure 8.25 shows that the parameter δ can still have a strong inﬂuence on the generalized viscosity for |d|2 < 0.1. For determining the yielded/unyielded regions we can introduce a criterion using the quantity |te | = ηgen |d| (we say the material has yielded when ηgen |d| > 2|β1 (E)|). As a consequence the yielded regions are inﬂuenced by the choice of δ if |d|2 < 0.1; for |d|2 > 0.1, δ does not change signiﬁcantly the shape of the yielded regions.

8.4 Applications: Channel Flow of ERFs

Fig. 8.24. ηgen (E = 2, |d|2 ) for diﬀerent nδ in the range 0.0001 < |d|2 < 0.01 (Casson-like model)

343

Fig. 8.25. ηgen (E = 2, |d|2 ) for diﬀerent nδ in the range 0.01 < |d|2 < 1 (Casson-like model)

Fig. 8.26. Dimensionless pressure (at x2 = 0) over the electrode ends for diﬀerent values of nδ ; the inset shows the detail of the pressure near x1 = 20

In Fig. 8.26 we plot the computed pressure evolution in the channel for diﬀerent values of nδ . In the region with constant electric ﬁeld (between the electrodes far from the electrode edge) the pressure drop is not inﬂuenced by nδ ; however, in the transition region nδ produces some diﬀerences. For values less than nδ used in the simulations (0.002) the diﬀerences are unnoticeable. In order to see the inﬂuence of δ on the ﬂow ﬁeld we plotted the velocities in the transition area (see Fig. 8.27) and in the middle of the electrodes (see Fig. 8.28) for diﬀerent values of nδ . It is seen again that for small values of nδ (nδ < 0.002), the proﬁles are almost congruent. Figures 8.27 and 8.28 permit in principle to identify those ﬂuids which are described by our model con-

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8 Electrorheological Fluids

Fig. 8.27. Proﬁles of v1 at x1 = −20 (electrode ends) for diﬀerent values of nδ

taining a larger nδ -value. For increasing nδ , the plateau is curving, indicating a reduced yielded region. Comparison with Experiment In order to compare the numerical modeling with the experimental results reported in [64] we considered in the numerical program a conﬁguration consisting of two electrodes, charged non-symmetrically, of length 2l = 40 mm, with a channel height of 2h = 2 mm and a length of the channel of 2L = 70 mm which corresponds to the geometry used in the experiments [64, 269]. Experiments are performed in a channel of width b = 20 mm. For computations the input value is the volumetric ﬂow rate in the channel Q = 43 hbv0 and the output is the dimensional pressure drop calculated from x1 = −30 mm on a length of 60 mm. In Fig. 8.29 we compare the obtained results for diﬀerent values of Eu = V /h. The quantity Eu represents the value of the uniform electric ﬁeld established in the middle of the channel, far from the electrode edges, and it is in fact the value of the magnitude of the electric ﬁeld if the electrodes were inﬁnitely long (see Sect. 8.4.2). While good agreement is obtained, the remaining inaccuracy of the calculations is given by the uncertainty in ﬁtting values for β1 and β2 (see (8.4.71) and (8.4.72)).

8.4 Applications: Channel Flow of ERFs

345

Fig. 8.28. Proﬁles of v1 at x1 = −16 (between the electrodes) for diﬀerent values of nδ

We considered this result as a validation of our computational model and used it to simulate further conﬁgurations in our search for a possible enhancement of the electrorheological eﬀect. b) Numerical Results – the Alternative Power-Law Model Material and Conﬁguration Properties Used for the Simulations To illustrate the power-law constitutive equation we chose the ER-ﬂuid EPS 3301 because, as shown in [269] it is described well by this model. EPS 3301 is produced by the company CONDEA Chemie AG (Germany). It is a particlefree ﬂuid and consists of an acid metal soap which is dissolved homogeneously in a conventional hydraulic basis liquid. It contains also some additional hydraulic oil additives [48, 182, 269]. The ﬂuid was studied experimentally by Wunderlich [269]. In order to better characterize the ﬂuid, it was found that a dilution with 40% (weight percent) white oil (Weiss¨ol) is appropriate, (see [15]). Unlike the ER-ﬂuid Rheobay 3565, the ER-ﬂuid EPS 3301 shows in the absence of the electric ﬁeld pseudoplastic behaviour (the shear viscosity decreases with increasing shear rate). Under the application of an electric ﬁeld, also pseudoplastic behaviour is observed. In [269] the power law model (8.3.33) was found the most appropriate model to describe the data. Fitting

346 8 Electrorheological Fluids

Fig. 8.29. Comparison of the numerical and experimental results (dimensional pressure-drop vs. volumetric ﬂow rate)

8.4 Applications: Channel Flow of ERFs

347

Table 8.3. Values of the parameters m and n of (8.3.33) for diﬀerent electric ﬁeld strengths kV ] E0 E [ mm

0

1

2

3

4

m [Pasn ]

0.4702

10.8695

23.0439

36.8244

67.0970

n [−]

0.8697

0.3923

0.3249

0.2921

0.2361

the experimental curves “stress vs. shear rate” (obtained with the rotational viscometer), the values of m and n were calculated (see Table 8.3). We used these values in our bidimensional model (8.4.67). First, by ﬁtting the values of f = m(v0 /h)n (see Table 8.3) and n we have to obtain their dependence on the dimensional electric ﬁeld E0 E: f (E0 E) = f1 + f2 (E0 E) + f3 (E0 E)2 , n(E0 E) = n1 ((E0 E)2 + n2 )n3 .

(8.4.76) (8.4.77)

The coeﬃcients f1 , f2 and f3 depend on the values of v0 and h. For v0 = 0.1647 m/s and h = 0.001 m, we obtained the values given in Table 8.4. The function f is written as a series expansion by retaining only terms up to second order while n is described by a power function. The experimental data (see Table 8.3) are provided for a limited range of the electric ﬁeld and do not cover the range of the electric ﬁeld employed in the simulations. We use formulas (8.4.76) and (8.4.77) also to extrapolate the given data for f and n for larger values of E. Figure 8.30 illustrates how the ﬁtted curves approximate the data and how the extrapolations for v0 /h = 164.7 s−1 look like. The geometry used for the simulation of EPS 3301 is the same as that used for Rheobay. The electric permittivity of EPS is ε2 = 1.4 × 10−10 A·s/(V·m), and we choose ε1 = 0.02 ε2 . The density of EPS 3301 is 850 kg/m3 . All simulations except those done to conduct the comparison with the experimental data from Fig. 8.44 are performed for h = 1 mm, E0 = 1 kV/mm, V = 2 kV and v0 = 0.1647 m/s. So, Re = 0.578 and Ma = 0.284. We considered here also nδ = 0.002. We will investigate later the inﬂuence of this parameter on the solution. Table 8.4. Values of the coeﬃcients fi and ni from (8.4.76), (8.4.77) for v0 = 0.1647 m/s and h = 0.001 m i

1

2

3

fi

41.431

33.284 mm/kV

2.958 (mm/kV)2

ni

0.398 (mm/kV)2−n3

0.008 (kV/mm)2

−0.163

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8 Electrorheological Fluids

The Flow Near the Electrode Ends (long electrodes) Here a similar analysis as for the alternative Casson-like ﬂuid is carried out. In order to study the inﬂuence of the electrode edges on the ﬂow we chose the same conﬁguration of the electrodes and boundary conditions for the electrical problem as for the ER-material Rheobay. Consequently, the diﬀerent rheological response must be due to the material properties. One can distinguish three regions: (1) one with small electric ﬁeld, far upstream of the entrance between the electrodes; (2) the transition zone with strongly inhomogeneous electric ﬁeld right between the edges and (3) the uniform electric ﬁeld zone between the electrodes, far downstream the edges. In Fig. 8.31 the velocity modulus is plotted. In the areas far ahead and far beyond the electrode edges, the x2 -component of the velocity is very small and does practically not inﬂuence the x1 velocity component. The end eﬀect is reduced and the velocity develops smoothly from the region with reduced electric ﬁeld to the region with constant electric ﬁeld. Cross-sectional cuts of the surface plot in these regions are presented in Figures 8.34 and 8.35. Figure 8.35 indicates the negligible value of v2 (∼ 1/1000 of v1 ). The proﬁle of v1 from the ﬁrst zone is close to a Newtonian proﬁle (reduced pseudoplastic behaviour as it was observed in the experiments for small electric ﬁelds in [269]). In the third zone the velocity proﬁle for constant electric ﬁeld presents the character of the unyielded region in the middle of the channel, as expected in general for an ER-ﬂuid. To quantify the ﬂow in the transition region we choose three points near the electrode ends: x1 = −20.2, x1 = −20 and x1 = −19.5 (see Figs. 8.32 and 8.15). Notice the smooth plateaux in the middle parts of the plots of v1 . The curvature change of v1 at x1 = −20 is very close to the electrode edge but only slightly visible and much less than in the corresponding plot of the alternative Casson-like model (middle plot of Fig. 8.14). The x2 component of the velocity is smaller than 5% relative of the maximum inlet velocity. As for the alternative Casson-like model, the two extreme values of v2 in x1 = −20.2 (ﬁrst positive and then negative) and x1 = −19.5 (ﬁrst negative and then positive) indicate that the ﬂuid avoids the electrode tips and goes around them. In order to study the inﬂuence of the electric ﬁeld on the pressure we plot in Fig. 8.36 the pressure along the channel, across the electrode ends (compare also with Fig. 8.18). The transition area is characterized by a higher pressure gradient compared with the pressure gradient in the region with small electric ﬁeld (outside the electrodes) and within the region with constant electric ﬁeld (between the electrodes, far from the edges). The diﬀerence between these two regions is not signiﬁcant. The presence of a higher gradient in the transition zone led us to the study of the possibilities to obtain a better ER-eﬀect. We continue our study of the transition region near the electrode ends with plots of the second invariant of the stretching tensor, |d|2 in Fig. 8.37 and of the generalized viscosity ηgen (8.4.67) in Fig. 8.38. Regarding the plot

8.4 Applications: Channel Flow of ERFs

349

Fig. 8.30. Fitting curves for the dependences of the coeﬃcients f and n on the electric ﬁeld. The insets show how the ﬁt is extrapolated

Fig. 8.31. Surface plot of the modulus of the non-dimensional velocity in the vicinity of the electrode edges (which are at x1 = −20)

of |d|2 against x1 , x2 , one can see that the regions with smaller values of |d|2 around the electrode edges are reduced as compared to the alternative Casson-like model (Fig. 8.19). This can be explained by the fact that the ER-ﬂuid EPS is a material with a reduced ER-eﬀect. However, the plot is qualitatively similar to the corresponding plot for Rheobay (Fig. 8.19). In Fig. 8.38 as in Fig. 8.20 a transition to a “solid” island between the electrode ends can be observed. Inﬂuence of the Parameter δ As for the alternative Casson-like model, δ can be viewed as a parameter which has to be chosen as small as possible to recover the pure power-law model (equations (8.3.51) and (8.3.52)) from equation (8.3.66). On the other hand, δ can be viewed as a constitutive parameter which can be obtained

350

8 Electrorheological Fluids

Fig. 8.32. Proﬁles of v1 at x1 = −20.2, −20, −19.5, respectively

Fig. 8.33. Proﬁles of v2 at x1 = −20.2, −20, −19.5, respectively

Fig. 8.34. Proﬁles of v1 at x1 = −23.5, −16.5, respectively

Fig. 8.35. Proﬁles of v2 at x1 = −23.5, −16.5, respectively

8.4 Applications: Channel Flow of ERFs

351

Fig. 8.36. Dimensionless pressure in the transition zone (x1 = −20 marks the electrode ends)

Fig. 8.37. Surface plot of dimensionless |d|2

from the experimental data when ﬁtting equation (8.3.66) with them. Here we treat it as a small parameter. The number nδ = δh2 /v02 is the dimensionless form of δ. Figures 8.39 and 8.40 show the inﬂuence of nδ on the dimensionless viscosity ηgen (E = 2, |d|2 ) according to formula (8.4.67) for diﬀerent ranges of the argument |d|2 . One can see that its inﬂuence becomes more drastic for small values of |d|2 . In

352

8 Electrorheological Fluids

Fig. 8.38. Surface plot of the generalized dimensionless viscosity

Fig. 8.39. ηgen (E = 2, |d|2 ) for diﬀerent values of nδ in the range 10−4 < |d|2 < 0.01 (alternative powerlaw model)

Fig. 8.40. ηgen (E = 2, |d|2 ) for diﬀerent values of nδ in the range 0.01 < |d|2 < 1 (alternative power-law model)

order to study the inﬂuence of nδ on the solution we plot in Fig. 8.41 (compare also Fig. 8.26) the pressure along the channel axis containing also the electrode ends for diﬀerent values of this parameter. Diﬀerences are only marginally visible for nδ < 0.002 while for larger values of nδ the curves are practically congruent. Similarly, the plots of the velocity proﬁles in the uniform electric ﬁeld and right between the electrode tips are shown in Figs. 8.42 and 8.43, respectively; they indicate a negligible inﬂuence of this parameter on the ﬂow.

8.4 Applications: Channel Flow of ERFs

353

Fig. 8.41. Dimensionless pressure (at x2 = 0) across the electrode end for diﬀerent values of nδ ; the inset shows the detail of the pressure near x1 = 20

Comparison with Experiment The experiments given in [269] have been performed for conﬁgurations consisting of two parallel electrodes charged non-symmetrically with three different lengths: 2l = 6 mm, 2l = 20 mm, and 2l = 40 mm. The channel height is 2h = 2 mm, the channel length is 2L = 70 mm and the channel width is b = 20 mm. The potential imposed on the upper electrode is V = 2 kV. The input value for computations is the volumetric ﬂow rate in the channel Q = 2(n + 1) v0 hb/(2n + 1). The output is the ER-eﬀect which is the dimensionless quantity F∆p (Q) =

∆pE (Q) −1, ∆pE=0 (Q)

(8.4.78)

where ∆p is the dimensional pressure drop calculated between x1 = −30 and x1 = 30 mm. In Fig. 8.44, the experimental results are compared with the numerical results for the three lengths of the electrodes. We found qualitative agreement: the ER-eﬀect is decreasing with increasing volumetric ﬂow rate and as expected, the ER-eﬀect is larger for larger electrode lengths l. The Femlab results are quantitatively close to the approximate analytic computations23 . This means that the eﬀect of the electrode ends is reduced, as is expected for a long electrode. 23

The electrorheological eﬀect (8.4.78) measured from x1 = −30 on a distance of 60 mm for the conﬁguration used in the experiments may be calculated annalytically (approximately) by using the procedure described in Subsect. 8.4.2

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8 Electrorheological Fluids

Fig. 8.42. Proﬁles of v1 at x1 = −20 (electrode ends) for diﬀerent values of nδ

c) Enhancing the ER-Eﬀect In technical applications there is a strong interest in obtaining an enhanced ER-eﬀect (8.4.78). Many experimental works (see e.g. [2, 269]) have been performed with a focus in this direction. Most attempts are based on the modiﬁcation of the geometry in such a way that inhomogeneities in the electric ﬁeld are introduced. These investigations show that changes in the geometrical conﬁgurations may lead to a reduction or an enhancement of the ER-eﬀect. However, it is diﬃcult to distinguish by what measure this enhancement is produced, i.e. by the electric ﬁeld inhomogeneities or by the inhomogeneities in the ﬂow which are due to the modiﬁcation of the geometry (which appear also when the electric ﬁeld is switched oﬀ). The conﬁgurations studied in this section are based on inhomogeneities caused by the end eﬀects of the electrodes. This means that instead of modifying the geometry, the electric ﬁeld inhomogeneities are introduced here by changing the boundary conditions in the electrical problem. Since by each such change, the formulation of the problem for vanishing electric ﬁeld remains unchanged, the increase/decrease in F∆p (see (8.4.78)) is equivalent to an increase/decrease in ∆p. Consequently, our study will focus on the examination of the possibilities to obtain larger values of the pressure drop at the same value of the volumetric ﬂow rate by using the inhomogeneities produced by the electrode ends. (by considering a uniform electric ﬁeld in the part of the channel between the electrodes and a vanishing electric ﬁeld in the remainder of the channel). One obtains F∆p = 2l(k − k0 )/(0.06k0 ) where k0 = (Q(2 + 1/n0 )/(hb))n0 m0 /h is the pressure gradient at a vanishing electric ﬁeld (see 8.4.49).

8.4 Applications: Channel Flow of ERFs

355

Fig. 8.43. Proﬁles of v1 at x1 = −16 (between the electrodes) for diﬀerent values of nδ

As Figs. 8.18 and 8.36 show, the pressure decreases more rapidly around the electrode edges than elsewhere. In order to analyse how one can use this eﬀect in an optimal way the simplest elements are investigated: short electrodes (which imply also an interaction between the left and right electrode edges), an electrode interrupted by a “hole” and a short electrode between two “holes”. The study is performed for both ER-materials Rheobay and EPS, which are modeled with the alternative Casson-like and power-law equations, respectively, as described previously. Since we treat here (almost) only cases with short electrodes, all approaches are numerical. The data introduced in Femlab are those given above (all lengths in this section are given relative to the channel height). Let us investigate ﬁrst the inﬂuence of the electrode length. To obtain the plot of Fig. 8.45, diﬀerent lengths 2l of the electrodes were considered, and the pressure drop ∆p = px1 =−30 − px1 =0 vs. l was computed. For l < 1.5 for the Casson-like ﬂuid and for l < 2 for the power-law ﬂuid, the curves (square points interpolation with continuous lines) deviate from the linear behaviour (solid straight lines) which is a consequence of the coupling of the electrode ends in the electrical problem. For very short electrodes (stronger inhomogeneities) the pressure drop is signiﬁcantly decreased. For the Casson-like ﬂuid, a better electrorheological eﬀect is obtained when the electrode edges do not interact while for the power-law ﬂuid the largest value of the ER-eﬀect is obtained for short electrodes of length about 0.5 where the electrode ends are strongly interacting. Note that 2 is the dimensionless height of the channel. The dotted lines of the same ﬁgures represent the pressure drop calculated

356 8 Electrorheological Fluids

Fig. 8.44. Comparison of the numerical and experimental results (electrorheological eﬀect vs. volumetric ﬂow rate)

8.4 Applications: Channel Flow of ERFs

357

Fig. 8.45. Pressure drop from x1 = −30 to x1 = 0 vs. the half length of the electrodes (Casson-like ﬂuid (left) and power-law ﬂuid (right)): dotted line – analytical evaluation with 1D model; square points interpolated with continuous line – numerical evaluation with 2D model; continuous straight line – ﬁt for the behaviour of the pressure drop for long electrodes

analytically by neglecting the end eﬀects. The diﬀerence between this curve and the curves obtained numerically is a measure of the values of additive pressure drops due to the end eﬀects. In order to see how the value of the pressure drop can be increased, we will investigate further what happens when the electrodes are interrupted by “holes” of length d (see Fig. 8.46). We study the eﬀect in two cases of charging the 4 electrodes: for V1 = V2 = V and for V1 = −V2 = V . We call these situations normal and inverse polarity of the electrodes, respectively. For inverse polarity, when the “hole” is suﬃciently short, there will appear a stronger electric ﬁeld that inﬂuences the ﬂow more signiﬁcantly. To illustrate this eﬀect we plot in Fig. 8.47 the velocity surface in the channel for the Casson-like ﬂuid. The behaviour for the power-law model is similar. The inhomogeneities in the ﬂow are stronger near the electrode edges bounding the “hole” than in the vicinity of the exterior edges. Of course, one should avoid too short “holes” which can lead to short circuits. We computed the average pressure gradient k1 = [p(x1 = −(l + d)/2) − p(x1 = (l + d)/2)]/(l + d) for l = 10 and for variable d. Since the electrodes are suﬃciently long, the electric ﬁeld becomes homogeneous and the ﬂow will be of Poiseuille type around x1 = ±(l + d)/2, the middles of the electrodes. By plotting k1 for diﬀerent lengths of the “hole” one can compare the “hole eﬀect” on the pressure drop. From Fig. 8.48 one can infer pretty similar qualitative behaviours for both ﬂuids. For d > 1 and normal polarity and for d > 2 and inverse polarity, the pressure gradient is smaller than that calculated for electrodes without any “hole”, k1 = 85.84 (for Rheobay) and k1 = 4.464 (for EPS) (corresponding to the case d = 0 for normal polarity). The behaviour for short “holes” is diﬀerent for the two polarity cases due to the diﬀerent interactions of

358

8 Electrorheological Fluids

the electrode edges in the “hole”: smaller values of the pressure gradient for normal polarity due to a reduced electric ﬁeld in the “hole” and higher pressure gradient for inverse polarity since a stronger electric ﬁeld occurs in this case in the “hole”. The optimal length of the “hole” is the shortest possible one without electric short circuit. If we introduce more “holes”, the next step is to study, for inverse polarity, how close the “holes” should be placed. To answer this question we study the case of one variable electrode of length l1 placed between 2 “holes” of constant length d (see Fig. 8.49). The electrodes on the left and right sides have length l2 . In Fig. 8.50 we plotted the average pressure gradient k2 = [p(x1 = −(l1 + l2 + 2d)/2) − p(x1 = (l1 + l2 + 2d)/2)]/(l1 + l2 + 2d) vs. l1

8.4 Applications: Channel Flow of ERFs

359

Fig. 8.49. Conﬁguration with electrodes between “holes” (inverse polarity)

Fig. 8.50. Average pressure gradient vs. length of the middle electrode (one electrode placed between two “holes”, alternating polarity) (Casson-like ﬂuid (left) and power-law ﬂuid (right))

for l2 = 10 and d = 1. The optimal length of an electrode between “holes” of length d = 1 is around l1 = 2 in the Casson-like case while in the power-law case it is around l1 = 1.5. Having the optimal parameters, one can now design optimal series of electrodes. As an example we consider a series of ﬁve electrode pairs and we compare the pressure over an interrupted electrode pair with the pressure over an uninterrupted electrode pair (see Fig. 8.51). One can see that up to 30% enhancement of the pressure drop can be obtained when combined optimized series of “holes” and electrodes are used. To get a better insight we display in Fig. 8.52 the generalized viscosity. For the Rheobay ﬂuid two regions with increased viscosity can be discerned: one along the walls and another one in the middle of the channel. The ﬁrst one is due to the high electric ﬁeld around the edges of the electrodes. If we had uninterrupted electrodes, we would have a continuous yielded region characterized by a high viscosity in the middle of the channel. But in the case presented here this region is split into islands of high viscosity. One observes a similar behaviour for the EPS ﬂuid but to a lower extent.

360

8 Electrorheological Fluids

Fig. 8.51. Comparison between the pressure over an uninterrupted electrode pair and the pressure over an optimized series of ﬁve electrode pairs (inverse polarity)

Fig. 8.52. Surface plot of the generalized viscosity for an optimized series of ﬁve electrode pairs (inverse polarity)

Conclusions In this subsection we focused on pressure drop simulations due to its importance for applications. Eﬀects of the “holes” and of the lengths of the electrodes have been computed that provide direct insight into the building blocks of typical experimental set-ups. The analysis has been carried out by taking into account the polarity of the electrodes which critically inﬂuences the distribution of the electric ﬁeld in the channel. A better eﬀect is obtained

8.4 Applications: Channel Flow of ERFs

361

by introducing “holes” in the electrodes and by alternating the polarity of the electrode cuts as depicted in Fig. 8.49 than taking continuous electrodes. We found optimal lengths for the electrodes and “holes” relative to the height of the channel. 8.4.4 Electrodes with Modiﬁed Shape and Position Relative to the Flow – Experimental Results and Discussion The inhomogeneities of the electric ﬁeld studied in Subsect. 8.4.3 were introduced by changing the boundary conditions in such a way that the upper and lower electrodes are parallel and placed along the walls of the channel. The geometric conﬁguration of the ﬂow remains unchanged. Another way to create non-uniform electric ﬁelds is to modify the shape or the position of the electrodes. Several experiments with diﬀerent ER-ﬂuids were performed in order to study the inﬂuence of the geometry of the electrodes on the ER response. Katsikopoulos and Zukoski [108] obtained an increase of the stress by a factor of 2 by using electrodes with grooves and ridges perpendicular to the direction of the ﬂow when compared with the case having smooth electrodes. On the other hand and quite contrary, the ER-response was the same as for smooth electrodes when the grooves of the electrodes were disposed in the direction of ﬂow. Otsubo’s experiments [166] show that the non-uniformity of the electric ﬁeld obtained by using electrodes of diﬀerent types – one coated with a striped pattern of conductive lines and another one with a honeycomb pattern of conductive lines – is responsible for an enhanced ER-eﬀect not achievable with smooth electrodes. These initiating studies and results were followed by further systematic works of Abu-Jdayil [2], Abu-Jdayil and Brunn [4, 5] and Wunderlich [269], whose results we brieﬂy describe in this subsection. All three authors try to ﬁnd how the ER-eﬀect can be increased by altering the geometry of the smooth electrodes. We focus in this presentation only on those experimental conﬁgurations which are connected with the theoretical modelling presented in this section, namely those performed on slit ﬂows in a stationary electric ﬁeld that can be reduced to two-dimensional models. The ER-ﬂuid used in [2, 4, 5] for the slit ﬂow, P 1723, is supplied by the company Robert Bosch GmbH (Germany). It is a transparent ﬂuid consisting of paraﬃn oil and silica gel with particle diameters from 5 to 30 nm, and a dispersed phase volume fraction of 30%. In the experiments performed by Wunderlich in [269] under a direct current (DC) the ER-ﬂuid EPS 3301 was used which was already described in Subsect. 8.4.3. We mention that in the experiments performed in [2, 4, 5] the length of the electrodes (irrespective of their shape and position), deﬁned as the distance between the electrode ends, is l = 200 mm. The measurements were performed with ﬁxed ∆p, where ∆p is the pressure drop determined in the middle of the electrodes pair over a length of l∆p = 150 mm. The volumetric

362

8 Electrorheological Fluids

ﬂow rate Q was obtained by integrating the measured velocity proﬁles as a function of ∆p. The quantitative measure of the ER-eﬀect is given by the eﬃciency factor (F ) deﬁned as the reduction in volumetric ﬂow rate at constant pressure drop & Q|E=0 − Q|E && F = . (8.4.79) & Q|E=0 ∆p In order to quantify the inﬂuence of the modiﬁed geometry of the electrodes on the ER-eﬀect in [2] the quantity & F && f1 = (8.4.80) Fu &∆p is formed, where Fu is the eﬃciency factor determined for smooth electrodes ﬂush with the boundary. In all experiments reported in [269] and mentioned here the length of the electrodes is l = 40 mm. The pressure drop over the electrodes, ∆p was measured over a distance of l∆p = 60 mm containing the electrodes, as a function of the volumetric ﬂow rate at certain constant values of the applied electric ﬁeld24 . Since l∆p > l one has to take into account also the distance between the channel plates which is h = 2 mm. In order to represent the variation of the ER-eﬀect and of the electrical power due to the modiﬁed geometry of the electrodes in [269] the quantities & & F∆p & , (8.4.81) f2 = F∆p,h=2 mm &Q and

& & & f3 = Pel,h=2 mm & Pel

,

(8.4.82)

Q

respectively, are chosen. F∆p is deﬁned in (8.4.78), while Pel represent the electrical power dissipated to maintain the constant electric potential on the electrodes. The reference values F∆p,h=2 mm and Pel,h=2 mm (in the denominators of (8.4.81) and (8.4.82), respectively) are evaluated for the conﬁguration with parallel smooth electrodes placed along the channel walls with Q = constant25 . 24

25

Although the electric ﬁeld is inhomogeneous the authors calculate for each conﬁguration a mean value of the electric ﬁeld in order to be able to compare the results obtained with diﬀerent conﬁgurations at the same value of the electric ﬁeld. It would be perhaps more appropriate to compare the results at the same value of the applied voltage on the electrodes which is a constant. Although the author does not mention it, we understand that the denominators as well as the corresponding numerators are evaluated for the same value of the electric ﬁeld. We make an alternative proposal at the end of this subsection.

8.4 Applications: Channel Flow of ERFs

363

Fig. 8.53. Conﬁgurations with electrodes separated by a gap smaller (left) and larger (right) than the channel height

Fig. 8.54. Conﬁgurations with convergent (left) and divergent (right) oblique electrodes

Let us make an important remark: the geometry of the electrodes may directly inﬂuence the ﬂow ﬁeld and beyond the inhomogeneity of the electric ﬁeld. In other words the ﬂow is non-viscometric due to the geometry even in the absence of the electric ﬁeld. In order to study what inﬂuence such a nonviscometric ﬂow may have on the ER-eﬀect, two conﬁgurations with parallel smooth electrodes with diﬀerent distances (see Fig. 8.53) were investigated in [269]. The ﬂow becomes non-viscometric at least in the vicinity of the edges and corners near the electrode ends. From the measurements with protruding electrodes for hel = 1 mm and hel = 0.5 mm, f2 < 1 and f3 < 1 were found. From the measurements with retreated electrodes for hel = 3 mm instead f2 > 1 and f3 > 1 was obtained. Another investigated conﬁguration consists of oblique electrodes (see Fig. 8.54). The idea behind this arrangement is to avoid as far as possible the presence of edges and corners and to examine the eﬀects when the distance between the electrodes varies which leads in this case to convergence/divergence of the stream lines. There are no measurements with this conﬁguration conducted for DC in [269]. The conﬁguration with convergent electrodes was studied in [2, 4, 5] for h = 3 mm and hmin = 2 mm. When evaluating f1 the author compared F with Fu , where Fu is the eﬃciency factor for a conﬁguration with smooth parallel electrodes separated by a gap

364

8 Electrorheological Fluids

Fig. 8.55. Conﬁguration with corrugated electrodes (asymmetric arrangement)

Fig. 8.56. Conﬁguration with corrugated electrodes (symmetric arrangement)

of hs = 2.5 mm26 . The result was f1 = 1 for U < 1.7 kV and f1 < 1 for U > 1.7 kV. The conﬁgurations presented now are based on electrodes with modiﬁed surfaces. The inhomogeneities of the electric ﬁeld and the non-viscometric eﬀects of the ﬂow are intensiﬁed in this way. In order to study the inﬂuence of the surface of the electrodes on the EReﬀect grooved electrodes were used (see Figs. 8.55 and 8.56). This shape can be viewed as a series of protruding and retreating electrodes. For d = 2 mm, h1 = 2.15 mm and h2 = 2.3 mm in [269] an increased ER-eﬀect f2 > 1 was obtained at approximately the same electric power f3 ≈ 1 for both symmetric and asymmetric arrangements. In [2, 4, 5] the dimensions characterizing the applied grooved electrodes are d = 10 mm, h1 = 2.5 mm and h2 = 3 mm. The eﬃciency factor F was compared with Fu taking the gap between the smooth parallel electrodes, hs = 2.3 mm. The results obtained were f1 > 1 for U < 1.7 kV and f1 < 1 for U > 1.7 kV. For U > 1.7 kV the eﬃciency factor for the asymmetric conﬁguration is slightly less than for the symmetric conﬁguration. In the above-described conﬁgurations the electric ﬁeld lines are approximately perpendicular on the stream lines. Wunderlich considered in [269] a conﬁguration (see Fig. 8.57) in which this behaviour is altered: the electrodes are shaped such that the electric ﬁeld lines are in some regions essentially parallel to the stream lines. Moreover, the ﬂow is obstructed by the beamed extrusions of the electrodes and the stream lines take a snake-like shape (the ﬂow direction will be alternatively horizontal and vertical). The ﬂuid ﬂows 26

This value is equal to the mean distance between the oblique electrodes. In this way, at a certain applied voltage U , the electric ﬁeld established between smooth electrodes, Es = U/hs is the same as the mean value of the electric ﬁeld between the oblique electrodes, Emean = U/hmean .

8.4 Applications: Channel Flow of ERFs

365

Fig. 8.57. Conﬁgurations with beamed electrodes

alternatively through regions with and without an electric ﬁeld. In spite of all these strong inhomogeneities the obtained ER-eﬀect is smaller than for the smooth electrodes: f2 < 1 and f3 < 1. The conclusion of all these experimental works is that an enhanced ER-eﬀect can be obtained when the inhomogeneities in the electric ﬁeld are introduced such that the ﬂow ﬁeld is perturbed as little as possible (i.e. remains essentially viscometric). The inhomogeneity in the ﬂow ﬁeld lowers the ER-eﬀect or at least cancels the gained ER-eﬀect due to the non-uniform electric ﬁeld so that the eﬀects compensate each other. In their approach the authors diﬀerentiate between two types of electrodes. In a conﬁguration based on the ﬁrst type the ﬂow becomes nonviscometric due to both the shape of the electrodes (presence of corners and edges) and the inhomogeneity of the electric ﬁeld. This means that the ﬂow is non-viscometric also when the electric ﬁeld vanishes. Conﬁgurations with protruding electrodes, retreated electrodes, beamed electrodes belong to this category. The second types of electrodes have shapes which are supposed to be perturbed only by a small measure of the viscometric ﬂow. Consequently, in this case, the non-viscometry of the ﬂow is due approximately to the non-uniform electric ﬁeld and in the absence of the electric ﬁeld the ﬂow is approximately viscometric. Oblique electrodes and corrugated electrodes are considered of this second type. However, since the mechanical problem is coupled with the electrical problem in a non-linear way, this separation can be used only as a guiding hint. A more rigorous approach would be accomplished by complementing the experimental studies by theoretical modeling. Concerning the theoretical approach to be applied to these complex conﬁgurations some remarks should be made. Thanks to its generality, the formulation of the problem given in Subsect. 8.4.1 permits its application and numerical implementation also for modiﬁed geometries of the channel (e.g. with oblique electrodes, grooved electrodes or with electrodes separated by a gap smaller/larger than the channel height), by simply adapting the respective boundary conditions. However this formulation is restricted to electrodes of zero thickness. Luckily, one can approximate each of the presented conﬁgurations with a corresponding one having electrodes of zero thickness by imposing the applied electric potential only on the interior boundaries of the electrodes. The main diﬃculty which arises in the numerical implementation is related to the choice of the mesh. A suﬃciently reﬁned mesh around the

366

8 Electrorheological Fluids

edges and corners requires huge memory space and excessively large CPU times to ensure convergence since at sharp electrode corners the electric ﬁeld is singular (see [246]). An important remark must be made about the empirical evaluation of the quantity E, which appears in (8.4.79)–(8.4.82), and it is approximated by the mean value of the electric ﬁeld. Nevertheless, the electric ﬁeld is inhomogeneous (at least in certain regions) in each conﬁguration investigated in these works so it is not reasonable to characterize it by a single mean value for the whole domain. Since the aim of the experimental and theoretical investigations is to obtain enhanced eﬀects at the same power consumption by modifying the morphology of the electrodes, perhaps it would be more appropriate to compare the obtained ER-eﬀects for the initial and modiﬁed conﬁgurations at the same electric power instead of at the same mean electric ﬁeld. So, in order to conduct a correct comparison between the EReﬀect with normal electrodes and the ER-eﬀect with modiﬁed electrodes we suggest to choose for each modiﬁed conﬁguration an electric voltage such that the electric power is the same as the power dissipated for the conﬁguration with smooth electrodes. This means for Wunderlich’s approach that & & Pel & = 1. (8.4.83) Pel,h=2 mm &Q

9 Appendix

9.1 Appendix A: On Objectivity In this Appendix we brieﬂy state how one can show what transformation properties the various electromagnetic ﬁeld variables introduced in the main body of this book enjoy. To this end, we shall use three and four-dimensional notation. Let xA be the (contravariant) four-vector (xi , t), consisting of the position vector xi (i = 1, 2, 3) of a particle and time t, and let xA = xA (xB ) be any C 1 -transformation (xi , t) → (xi , t ). A covariant four-tensor ψAB and a contravariant four-tensor ψ AB are then quantities, which under such mappings transform according to ψAB =

∂xC ∂xD ψCD , ∂xA ∂xB

ψ AB =

∂xA ∂xB CD ψ . ∂xC ∂xD

(9.1.1)

Likewise, a contravariant four-vector transforms under general transformations (xi , t) → (xi , t ) according to σ A =

∂xA B σ . ∂xB

(9.1.2)

Of special interest are Euclidian transformations given by xi = Oij (t)xj + ci (t) ,

xi = Oji (t)(xj − cj (t)) ,

t = t ,

t = t .

(9.1.3)

In what follows we would like to explore some consequences implied by them. (i) Let σ A = (Ji , Q) be a contravariant vector. Then a routine calculation shows that under Euclidian transformations σ A = (Ji , Q ) = (Oij (Jj − x˙ j Q) + Qx˙ i , Q) ,

(9.1.4)

or K. Hutter et al.: Electromagnetic Field Matter Interaction in Thermoelastic Solids and Viscous Fluids, Lect. Notes Phys. 710, 367–374 (2006) c Springer-Verlag Berlin Heidelberg 2006 DOI 10.1007/3-540-37240-7 9

368

9 Appendix

(Ji − Qx˙ i ) = Oij (t)(Ji − Qx˙ j ) , Q = Q .

(9.1.5)

In other words, Q and (Ji − Qx˙ i ) = Ji transform under Euclidian transformations as an objective scalar and an objective vector, respectively. (ii) Let ψAB be a skew-symmetric covariant four-tensor with the components   0 b3 −b2 e1  0 b1 e2   (9.1.6) ψAB =  (−) 0 e3  0 (we choose to name these components bi and ei for suggestive reasons lateron). If we perform a Euclidian transformation, (9.1.3) shows that the three-vectors bi : (b1 , b2 , b3 ) and ei := (e1 , e2 , e3 ) transform as follows: bi = det (O)Oik bk , (9.1.7) ei + eijkx˙j bk = Oij (t)(ej + ejkl x˙ k bl ) . Otherwise stated, bi is an objective axial vector and (ei + eijk x˙ j bk ) an objective polar vector under the Euclidian transformation group. To prove (9.1.7)1 for instance, note that ψij = Oik Ojl ψkl ,

where

ψkl = eklm bm ,

where eijk is the usual three-dimensional permutation tensor. In much the same, though more complicated way, one can also prove that ei = Oil {el + elmn [O˙ jm (xj − cj ) − Ojm c˙j ]bn } , which, with the aid of the identity O˙ jm (xj − cj ) − Ojm c˙j = x˙ m − Ojm x˙ j , immediately implies (9.1.7). A special application of (9.1.6) is the tensor whose components are bi := mi , ei = 0. Then mi must be an objective axial tensor under the Euclidian transformation group. (iii) Another covariant skew-symmetric tensor of importance is   ˙ 1 0 m3 −m2 (m × x)  ˙ 2 0 m1 (m × x)  . ψAB =  (9.1.8)  (−) ˙ 3 0 (m × x) 0 It can be shown by a straightforward calculation, that the three-vector mi := (m1 , m2 , m3 ) is an objective axial vector under Euclidian transformations. (This is just a special case of (ii).)

9.1 Appendix A: On Objectivity

369

(iv) Let ψ AB be a skew-symmetric contravariant four-tensor with the components   0 h3 −h2 −d1  0 h1 −d2  . ψ AB =  (9.1.9)  (−) 0 −d3  0 A calculation identical to that performed above shows that the vectors d1 := (d1 , d2 , d3 ) and h1 := (h1 , h2 , h3 ) transform under the Euclidian group as di = Oij dj , hi − eijkx˙j dk = det (O)Oij (hj − ejkl x˙ k dl ) .

(9.1.10)

Hence di and (hi − eijk x˙ j dk ) are an objective vector and an objective axial vector under Euclidian transformations. A special situation is again the case for which h i = mi

and

di = 0

which immediately shows that mi must be an objective axial vector. (v) As a last example, consider the contravariant skew-symmetric tensor   ˙ 3 −(p × x) ˙ 2 p1 0 (p × x)  ˙ 1 p2  0 (p × x) . ψ AB =  (9.1.11)  0 p3  (−) 0 Its transformation properties are most easily found in two steps. Firstly, we write ∂xk ∂x4 l4 (ψ )k4 = ψ , ∂xl ∂x4 and obtain with the aid of (9.1.11) and (9.1.3)1 , pi = Oij pj ,

(9.1.12)

proving that pi is an objective vector. On the other hand (ψ )ij = pix˙j − pjx˙i =

∂xi ∂xj kl ∂xi ∂xj 4l ∂xi ∂xj k4 ψ + ψ + ψ , ∂xk ∂xl ∂x4 ∂xl ∂xk ∂x4

and it is now an easy matter, using (9.1.11) and (9.1.3), to show that the expression on the far right and in the middle of this chain are the same if pi is assumed to obey (9.1.12). It is shown in theoretical electrodynamics that the Maxwell equations of deformable continua can be written in the form

370

9 Appendix

eABCD

∂ϕCD =0 ∂xB

and

∂η AB = σA , ∂xB

(9.1.13)

where ϕCD and η CD are skew-symmetric covariant and contravariant fourtensors, respectively, and where σ A is a contravariant vector. Furthermore, eABCD is the four-dimensional permutation tensor, which is anti-symmetric with respect to any interchange of two indices and vanishes if any two indices are the same. Moreover, e1234 = 1, and lowering and rising of indices is achieved by the use of the metric tensor gAB = gAB , whose matrix is given by   100 0 0 1 0 0   gAB =  (9.1.14) 0 0 1 0  . 0 0 0 −1 Hence e1234 = −1, since e1234 = +1. The equations (9.1.13) are general and hold in vacuo as well as in matter, but the contribution of matter is usually separated from that of vacuo; this separation is achieved by writing ϕAB = ΦAB − µAB ,

η AB = H AB − π AB ,

(9.1.15)

where µAB and π AB are a covariant and a contravariant skew-symmetric four-tensor, respectively, which vanish in vacuo. Thus ΦAB and H AB are the vacuum ﬁelds. Note that in view of the transformation properties explained under (i)–(v) for general skew-symmetric tensors, there will be no need to derive such properties for ϕAB , ΦAB , µAB , η AB , H AB and π AB anew. Before we list these tensors in the various descriptions, recall that the vacuum-ﬁelds ΦAB and H AB are related to each other through the equation ˆAB (H CD ) , ΦAB = Φ

(9.1.16)

a relation which is sometimes called the Maxwell–Lorentz-aether relation. We shall see that it is not invariant under the general transformation (xi , t) → (xi , t ). We now list the various formulations and give the invariance properties which their variables enjoy. (a) Minkowski formulation. In this formulation one chooses µAB = 0 and does not separate η AB into two parts. Thus     0 B3 −B2 E1 0 H3 −H2 −D1   0 B1 E2  0 H1 −D2   , η AB =   . (9.1.17) ϕAB =    (−) 0  E3 (−) 0 −D3  0 0 Hence, because of the properties (ii) and (iv) of skew-symmetric tensors, we have under the Euclidian transformation group

9.1 Appendix A: On Objectivity

Di , Ei := Ei + eijkx˙j Bk , Bi , Hi := Hi −

eijkx˙j Dk

371

transform as objective polar vectors , ,

transform as objective axial vectors .

It is also a routine matter to show that (9.1.13) agrees with (3.3.36). Finally, the Maxwell–Lorentz–aether relation is formally introduced by writing η AB = H AB − π AB with     0 H3a −H2a −D1a 0 −M3 M2 P1       0 H1a −D2a  0 −M1 P2      AB = HAB =  , π  , a   (−) 0  −D (−) 0 P 3 3    0 0 (9.1.18) where Hia and Dia are auxiliary ﬁelds and Mi and Pi are the Minkowskian magnetization and polarization. Again, in view of items (ii) and (iv) above Dia , Pi , Mi := Mi − eijk x˙ j Pk Hi := Hia − eijk x˙ j Dk

transform as objective polar vectors , transform as objective axial vectors

(9.1.19)

under the Euclidian transformation group. Now, the Maxwell Lorentz– aether relation (9.1.16) is given by Hia =

1 Bi µ0

and Dia = ε0 Ei ,

(9.1.20)

and it is trivial to show that these are not invariant under Euclidian transformations. (b) Statistical Model. Except for notation this model is identical with the Minkowski model. Hence, under Euclidian transformations Dia , Ei , Pi , Mi , Bi , Hi ,

transform as objective polar vectors , transform as objective axial vectors .

(9.1.21)

(c) Lorentz formulation. In this formulation one sets µAB = 0, as was done in the previous formulations. Furthermore, H AB is given as in (9.1.18), but   ˙ 3 M2L − (P L × x) ˙ 2 P1L 0 −M3L + (P L × x)    ˙ 1 P2L  0 −M1L + (P L × x)   AB η =  . (9.1.22) L  0 P 3   (−) 0 This tensor can easily be written as the sum of two tensors, one containing the polarization, the other containing the magnetization only. From iv) and v) and the previous results it then follows that

372

9 Appendix

Dia , Ei , PiL = Pi , MiL = Mi , Bi , Hia ,

transform as objective polar vectors , transform as objective axial vectors .

under the Euclidian transformation group. (d) Two–Dipole Model (Chu formulation). This model is the only one with nonvanishing µAB . Indeed,     ˙ 1 0 −M3C M2C (M C × x) 0 B3a −B2a E1C       0 B1a E2C  ˙ 2 0 −M1C (M C × x)     ΦAB =   , µAB =   C  C   0 E3  ˙ 3 0 (M × x)   (−)  (−) 0 0     ˙ 3 −(P C × x) ˙ 2 P1C 0 H3C −H2C −D1a 0 (P C × x)     C C   0 H1C −D2a  ˙ 0 (P × x) P 1 2     η AB =   .  , π AB =  a C  (−)  0 −D3  0 P3     0

(−)

0 (9.1.23)

Here, all variables are the so-called Chu-variables and Bia = µ0 HiC

and

Dia = ε0 EiC

are the Maxwell–Lorentz–aether relations. It follows from (ii)–(v) above that under Euclidian transformations Dia , Ei := Ei + eijk x˙ j Bka , PiC , Bia , Hi

:=

HiC

−

eijk x˙ j Dka , MiC

transform as objective polar vectors , ,

transform as objective axial vectors .

It is not diﬃcult to show for each set of four-tensors, introduced above that the three-dimensional Maxwell equations in the respective formulations are obtained. Moreover, as can be clearly seen from the above derivation, Ei (or EiC ) and Hi (or Hia or HiC ) are not objective vectors under the Euclidian transformation group but that ε0 Ei , ε0 EiC , µ0 Hi , µ0 Hia and µ0 HiC are, as can easily be seen by invoking the Maxwell–Lorentz–aether relations in the expressions for Dia , Bia . Note also, that MiL and MiC are both objective. Of such properties we have freely made use in the main body of this book. Finally, we mention once more that it is through the Maxwell– Lorentz–aether relations that the Maxwell equations are not invariant under general transformations (xi , t) → (xi , t ). Their form is such that the Maxwell equations in vacuo (in which the Maxwell–Lorentz–aether relations are substituted) are invariant only under a very restricted transformation group, the Lorentz group. This should not be confused with the basic fact that ϕAB , ΦAB , µAB , η AB , H AB and π AB are four-tensors, which must obey (9.1.1). Consequently, the transformation properties under

9.2 Appendix B: Some Detailed Calculationsof the Maxwell–Minkowski Model

373

the Euclidian group hold irrespective of the invariance properties of the Maxwell equations. Of course, Euclidian transformations are one group only, for which the transformations (9.1.1) and (9.1.2) hold. In principle, any other transformation group can be investigated and of special interest are Lorentz transformations, because they leave the Maxwell equations including the Maxwell– Lorentz–aether relations invariant. These transformations are well-known and so we do not elaborate on them.

9.2 Appendix B: Some Detailed Calculations of the Maxwell–Minkowski Model In this Appendix we present a motivation for the choice of ρT and R, (3.4.18), and a derivation of equation (3.4.20). We start by transforming the Poynting-vector EM × H M . With the aid of (3.4.1) we show that eijk EjM HjM = eijk (Ej − ejlm x˙ l Bm )(Hk + ekpq x˙ p Dq ) = eijk Ej Hk + (Ej Dj + Hj Bj )x˙ i − (Ej Di + Hj Bi )x˙ j +ejkl Pk Bl x˙ j x˙ i + O(V 2 /c2 ) .

(9.2.1)

Next, using the Maxwell equations (3.4.10) and (3.4.13), we derive

(eijk Ej Hk ),i = −Ji Ei − Di Ei − B i Hi ˙i = −Ji Ei − (ε0 Ei E˙i + µ0 Hi H˙ i ) − Ei P˙ i − µ0 Hi M −(Ej Dj − Hj Bj )x˙ i,i + (Ei Dj + Hi Bj )x˙ i,j (9.2.2) d 1 ˙i − µ0 Hi M ˙i P = −Ji Ei − (ε E E + µ H H ) − E 0 i i 0 i i i dt 2 −(Ei Pj + µ0 Hi Mj )x˙ i,j − (ε0 Ej Ej + µ0 Hj Hj )x˙ i,i +(Ei Dj + Hi Bj )x˙ i,j . Substitution of (9.2.1) and (9.2.2) into (3.4.17) yields (ρ˙ + ρx˙ i,i )(U + 12 x˙ i x˙ i + T ) + ρT˙ − ρU˙ + qi,i − ρrext

−Ji Ei − Ei P i −µ0 Hi Mi −[Ri − (Pj Ej + µ0 Hj Mj )x˙ i +ejkl Pk Bl x˙ j x˙ i ],i − [tij + Ei Pj + µ0 Hi Mj ]x˙ i,j +[ρ¨ xi −

ρFiext

+ 12 (ε0 Ek Ek

− tij,j − (Ei Dj + Hi Bj ),j

+ µ0 Hk Hk ),i ]x˙ i = 0 .

(9.2.3)

374

9 Appendix

For this relation to be invariant under rigid-body translations, the term proportional to x˙ i x˙ j , i.e.: ejkl Pk Bl x˙ i x˙ j , must be compensated by Ri . Moreover, R must vanish with vanishing P and M . Both requirements are satisﬁed by assuming (3.4.18)2 which reads Ri = (Pj Ej + µ0 Mj Hj )x˙ i + ejkl Pk Bl x˙ j x˙ i .

(9.2.4)

Furthermore, if T would not be an objective scalar under rigid-body translations (with velocity b(t)) the term ρT would lead to a term proportional to ˙ b(t). Since this would be the only term of this kind, the above assumption leads to a contradiction and, hence, ρT must be objective. In that case, there is no distinction possible between T and U , and so T may be absorbed by U , or in other words, we may take ρT = 0 ,

(9.2.5)

as was done in (3.4.18)1 . We now substitute (9.2.4) and (9.2.5) into (9.2.3) and use the relation (Ei Dj + Hi Bj ),j − 12 (ε0 Ek Ek + µ0 Hk Hk ),i

= QEi + eijk Jj Bk + Pj Ej,i + µ0 Mj Hj,i + eijk (Dj B k + Dj Bk ) , which follows from (3.4.10) and (3.4.13). This then leads to (3.4.20).

(9.2.6)

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Name Index

Abu-Jdayil, 281, 362 Adler, 13, 39, 52, 84, 87 Alblas, X, 24, 55, 65, 85–88, 103, 150, 197, 202, 209 Alers, 196 Ambartsumian, 203, 257 Amp`ere, 13, 72, 85, 87, 133, 197, 223, 224, 238, 269, 272 Atkin, 330 Bagdasarian, 203, 257 Barnes, 314 Bast, 203 Baumhauer, 150 Benach, 71, 73 Bernoulli, 3, 214, 220, 232 Bessel, XVIII, 219, 260, 264 Bingham, XXIII, 4, 309–318, 327, 328 Biot, XV, XVI, 3, 201, 202, 229–231, 233, 235–239, 242–244, 253 Boﬃ, 66 Borghesani, 82, 83, 168 Boulanger, X, 87 Brown, IX, 51, 56, 85, 279 Brunn, 281, 362 Carath´eodory, 49 Casson, XXI, XXIII, 283, 307, 309–311, 313, 314, 317, 318, 324, 327, 328, 331, 333–335, 338, 340–344, 351, 352, 357–360 Cauchy, XV, XX, 8, 10, 28, 34, 134, 137, 272, 283, 311, 323 Chadwick, 12 Chang, 257 Chattopadhyay, 197, 202, 203, 271, 275 Chen, 5

Chu, XVI–XIX, 13, 39–41, 46, 51, 55, 83, 84, 87, 88, 91, 92, 103, 104, 106, 120, 125, 132–134, 137, 145, 157, 171, 173, 174, 179, 182, 189, 374 Civit` a, 12 Clausius, 305, 312 Coleman, 29, 47 Collet, 87 Couette, 312, 318, 329 Couwenberg, 203 Dalrymple, 197, 203, 216 De Groot, 24, 55, 66, 67, 71 Ding, 5 Dirichlet, 321, 331 Dixon, 86, 88 Duhem, 54, 295, 305, 312 Dunkin, 196 Eckart, V, VI, 282, 285, 287, 302, 304, 307, 324, 325, 328, 330 Eringen, 5, 6, 13, 24, 86–88, 98, 150, 196, 202 Ettinghausen, 82 Euclid, 7, 14, 16, 21–28, 33, 40, 41, 44–46, 48, 52, 53, 56–60, 68, 70, 73, 76, 78, 84–86, 92, 106, 127, 369–375 Euler, XVI, XVII, XIX, XXI–XXIII, 8, 9, 14, 103, 104, 107, 117, 119, 121–123, 130, 138, 148, 150, 194, 196, 197 Fano, 13, 39, 51, 84, 87 Faraday, 11, 152, 173 Feynman, 67 Finger, XVI

392

Name Index

Fleary, 196 Fourier, 79, 82, 83, 167, 264 Galilei, 21–27, 33, 41, 70 Gauss, 11, 55, 56, 87, 99, 152, 157, 173, 226 Gibbs, 49–51, 54, 63, 70, 77, 111, 114, 116, 117, 124, 125, 129, 152 Glen, 317 Goudjo, 202 Green, XV–XVII, 8, 28, 55, 86 Grot, 86, 98 Hamilton, 87, 261, 276 Hara, 203 Hasanyan, 5, 257 Haus, IX, 11, 13, 39, 42, 63, 64, 66, 72, 84, 85, 87, 92, 98, 133 Helmholtz, XXI, XXIII, 3, 47–49, 54, 70, 75, 77, 89, 109, 111, 115, 124, 129, 137, 208, 210, 295, 299 Hhefni, 6 Hilbert, 221 Hild, 317 Hooke, 225, 226, 258, 273 Hopf, 330 Hsieh, VI, 5 Hutter, V, IX, X, 12, 24, 29, 40, 42, 44, 51, 54, 55, 71, 73, 76, 84, 87, 103, 104, 117, 123, 138, 149, 150, 196, 317 Ikeda, 5 Ishmukhametov, 87 J¨ ohnk, 12 Jacobi, XVII Jang, 6 Jiang, 5 Jordan, 150 Kaliski, 196 Kamlah, 5 Karlash, 257 Katsikopoulos, 362 Kazarian, 203 Kirchhoﬀ, XX, 15, 109, 110, 114, 124, 125, 136, 163, 165, 166, 259 Kronecker, XXI, 8

Kumar, 257 Lagrange, XV–XIX, XXII, XXIII, 8, 14, 16, 17, 36, 103, 104, 106–111, 117, 119, 121, 126, 127, 129, 132–134, 137, 138, 141, 148, 150, 152, 157, 169, 171, 193, 194, 212, 213, 223, 239, 261, 276, 312, 336 Lam´e, XXII, 188 Lancioni, 6 Landau, 285 Laplace, 3, 208, 260, 321 Legendre, XXII, XXIII, 50, 77, 89, 116, 125, 182, 190, 223 Levi, 12 Li, 5 Liapunov, 197 Librescu, 257 Lieshout, 204, 214, 254 Lifschitz, 285 Liu, 24, 71, 73, 76, 257 Lorentz, XV, XVI, XVIII, XIX, XXI, XXII, 21, 22, 24–26, 33, 36, 41, 42, 44–46, 52, 53, 57, 58, 66, 68, 71, 72, 77, 84, 86, 87, 96, 98, 103, 105, 107, 108, 117–121, 123–125, 130, 132–135, 137, 145, 152, 168, 170–173, 176, 182, 189, 194, 197, 223, 229, 268, 269, 271, 372–375 Love, 259 M¨ uller, X, 22, 24, 29, 55, 71, 73, 76, 295 Marusewski, VI Mason, XIX, 332, 333 Mathieu, 209 Maugin, VI, 5, 6, 13, 24, 87, 88, 202 Maxwell, XV–XXI, 4, 7, 13, 16, 19, 21, 23–25, 29, 33, 36, 37, 39, 41, 45–47, 52, 56, 58, 60, 63, 65, 66, 71, 72, 83–86, 90, 92, 93, 95, 97, 103–108, 117, 118, 120, 125–127, 132–134, 136, 150, 153, 156, 157, 160–162, 171–175, 189, 197, 207, 213, 224, 226, 258, 259, 268, 269, 271, 272, 284, 285, 287, 288, 292, 293, 295, 297, 298, 320, 321, 330, 371–375 Mayn´e, 87 McCarthy, 196

Name Index Mellgren, 315 Minkowski, 4, 14, 33, 56–58, 63, 65–67, 92, 93, 95, 103, 117, 118, 121, 125, 132–134, 171, 173, 175, 189, 197, 207, 213, 224, 226, 258, 259, 268, 269, 271, 272, 284, 285, 287, 288, 292, 293, 295, 297, 298, 320, 321, 372, 373 Miya, VI, 5, 202–204, 217 Mol’chenko, 203 Moon, VI, 196, 197, 202, 203, 206, 209, 210, 216, 217, 229, 233, 257, 269 Morro, 82, 83, 168 Navier, 214 Nemoto, 204 Nernst, 82 Newton, 305, 307, 310–314, 317, 327, 328, 332, 334, 338, 340, 351 Noll, 22, 28, 29, 47 Nowacki, VI, 203 O’Peach, 203 Ohm, 82, 83, 167, 304 Onsager, 79, 82, 83, 131, 132, 154 Otomo, 203 Otsubo, 362 Ovakimian, 203 Pao, X, 6, 13, 24, 40, 42, 44, 51, 65, 71, 84–87, 138, 149, 150, 196, 197, 202, 206, 210, 216, 217 Papanastasiou, 314–316 Paria, 196 Parkus, X, 6, 87 Peach, 197 Peltier, 82 Penﬁeld, IX, 11, 13, 39, 42, 63, 64, 66, 72, 84, 85, 87, 92, 98, 133 Petykiewicz, 196 Pfaﬀ, 49 Piliposian, 5 Piola, XX, 15, 109, 110, 114, 124, 125, 136, 163, 165, 166 Pipkin, 82, 168 Podil’chuck, 5 Poiseuille, V, 4, 359 Poisson, XXII, 238, 321 Popelar, 197, 203

393

Poynting, 287 Prechtl, X, 88, 103 Qin, 5 Rajagopal, VI, 281, 305, 307, 330 Rayleigh, 202, 258, 261, 262, 276 Reynolds, XX, 11, 332, 333 Rivlin, 55, 82, 86, 168 Romano, 88 Romeo, 6 Rongen, 204 Rosenfeldt, 66, 71 R˚ uˇziˇcka, VI, 306, 307, 330 Sadiki, 282 Savard, XV, XVI, 3, 201, 202, 229–231, 233, 235–239, 242–244, 253 Savart, 229 Seebeck, 82 Shindo, VI Sindo, 5 Smits, 204 Stokes, 11 Strouhal, 289 Suttorp, 24, 55, 66, 67, 71 Takagi, 204 Tanaka, 256 Tani, VI, 203 Taylor, 5, 42, 139, 140, 143, 146, 163, 185 Thomson, 82 Tiersten, 5, 13, 85, 86, 150, 197 Tiesten, 24 Timoshenko, 264 Tomasetti, 6 Toupin, 13, 24, 40, 51, 71, 73, 85–87, 103, 138, 149 Truesdell, 13, 28, 40, 71, 73 Tsai, 86 Ursescu, V Van de Ven, V, X, 24, 55, 59, 87, 138, 149, 197, 202–204, 207, 209, 214, 254, 264, 266 Van Geen, 87 Vischer, X

394

Name Index

Vlasov, 87 Vliegelahn, 197 von-K´ arm´ an, 257 Wallerstein, 197, 203 Wang, 5 Wiener, 330 Wineman, 305 Winslow, 280 Wolfe, 203 Wunderlich, 328, 362, 363, 367

Yabuno, 203 Yang, 5, 202 Yao, 256 Yeh, 85, 150, 197 Young, XVI, 238

Zheng, 202, 204 Zhou, VI, 5, 6, 13, 24, 202, 204 Zukoski, 362

Subject Index

absolute temperature, XXII action functional, XVIII action integral, XVIII ﬁrst variation, XXI second variation, XXI actuator, 282 Amp`erean current model, 85, 197 angular momentum equation, 127 angular velocity, XXIII anisotropy coeﬃcient, 145 augmented Lagrangean, 312 auxiliary ﬁeld, 171 balance law energy, 9, 29, 73, 153, 293 internal energy, 283 mass, 9, 29, 73, 283, 293 momentum, 9, 73, 90, 153, 283, 293 angular, 29 linear, 29 momentum of momentum, 9, 110, 283 total energy, 294 balance law of electrorheology, 293 balance law of mechanics, 9 beam, XIX, 2 beam equation, 218, 230 beamed electrode, 366 bending energy beam, XXI plate, XXI bending moment, XVII bending theory, 220 Bernoulli beam, 3 Bernoulli–Navier theory, 214 Bessel function, XVIII, 219 bi-viscosity model, 314 bidimensional, 346

Bingham behaviour, 312, 327 Bingham ﬂuid, 328 Bingham like model, 318 Bingham model, 310, 311, 314, 316, 317 alternative, 315, 316 Biot–Savard, 3 approach, 3 force, XVI Biot–Savard ﬁeld, 239 Biot–Savard method, 201, 233, 235, 238, 242–244, 253 direct, 236 body force electromagnetic, XVI external, XVI Brownian motion, 279 buckling, 197 buckling current, 227, 233, 237, 238 buckling ﬁeld, 213, 266 magnetic, 214, 220 buckling mode, 228 anti-symmetric, 219, 242 symmetric, 219 buckling value, 2, 206, 219, 221, 222, 261 cantilevered beam, 201 cantilevered ferromagnetic beam, 206 carrier ﬂuid, 280 Casson, XXI Casson behaviour, 327 Casson ﬂuid, 328 Casson model, 307, 314, 318 extended, 311 Casson-like constitutive function, 314 Casson-like ﬂuid, 340, 341, 357–359 Casson-like model, 283, 310, 314, 318, 324, 327, 334, 342, 351

396

Subject Index

alternative, 332, 334 Casson-like viscosity, 343 Cauchy stress tensor, 283, 304 Cauchy–Green deformation tensor, XV, 8 channel ﬂow, 318 charge density, XIX electric, XIX four-vector, XXII in the rest frame, XIX magnetic, XIX surface, XIX Chu formulation, 39, 104, 134, 145, 171, 174, 374 Chu model, 91, 134 circular beam, 251 Clausius–Duhem inequality, 305, 312 clutch, 282 combined (variational Biot-Savard) method, 238 combined method, 202, 236, 244 composite particle, 279 concentric superconducting ring, 231, 277 concentric tori, 232 conductive current, XVII, 12 in the rest frame, XVIII conductor ﬂat spiral, 250 helical, 236 ﬁnite, 237 inﬁnite, 237 inﬁnite one-dimensional, 240 spiral, 236 conﬁguration present, 7 reference, 7, 36 conservation charge, 12 magnetic ﬂux, 11 constant dielectric, XXII elastic, XV, 145, 196 electric, 195 electric anisotropy, 195 electromagnetic coupling, 195 electrostrictive, XV, 145 Lam´e, XXII, 188

magnetic, 195 magnetic anisotropy, 195 magnetostrictive, XV, 145 piezoelectric, XXII, 195 piezomagnetic, XXII, 195 pyroelectric, XXII, 195 pyromagnetic, XXII, 195 thermal, XV, 195 thermoelastic, XXII, 145, 195 thermoelectric, XVIII, 145, 195 thermomagnetic, XVIII, 145, 195 constitutive equation, 26, 77, 116, 298 framework, 283 postulate, 91 relation, 130, 145, 153 contravariant four-tensor, XXIII, 371 convective derivative, XXIII, 290 corotational, XXIII Couette ﬂow, 312 Couette-type, 329 couple supply, 52 covariant four-tensor, XXIII, 370 cross-section elliptic, XV, 3, 201, 207, 209, 216 rectangular, 201 current density, 106 conductive, XVIII electric, XVIII magnetic, XVIII non-conductive, XVIII electric, XVIII magnetic, XVIII cylindrical coordinate, 234 damper, 282 decomposition of the balance laws, 156 of the constitutive equations, 163 of the jump and boundary conditions, 169 of the momentum and energy equations, 159 deformation gradient, 8 dielectric assumption, 288 dielectric displacement, XVI, 12 auxiliary, XVI in the rest frame, XVI Lagrangean, XVI

Subject Index discharge, XX displacement, 242, 259, 272, 273 from the initial to the intermediate state, XXII domain interaction, 281 dual tensor of the body couple vector, XVIII Duhem relation, 54 eﬀective conductivity, 302 eﬀective magnetization, 284, 288 eﬀective permittivity, XXI, 302 eigenfrequency, 256, 275, 277 eigenvalue, 208, 253–255 eigenvector, 253 elastic bending moment, 269 elastic energy, 226, 232, 233, 251 elastic stress, 258 elasticity tensor, 225 electric charge, 320 circuit, 36 conductivity, XXII eﬀective, XXII current, XXII, 77, 130 current ﬁeld, XVII dipole, 36 displacement, 1 ﬁeld strength, XVI in the rest frame, XVI ﬁeld-magnetic induction four-tensor, XXIII potential, XXI susceptibility, 188 second- and fourth-order, 145 electric non-conducting ﬂuid, 321 electrical conduction, XV conductor, 79, 131 insulator, 78, 130 electrical model of order zero, 304 electrically compressible, 305 electrically conducting ﬂuid, 301, 320 electrically non-conducting ﬂuid, 302 electrode edge, 331, 340 electrode end, 336, 349 electrode of ﬁnite length, 329 electrode with modiﬁed shape, 361 electrohydrodynamic convection, 281

397

electromagnetic body force, 52, 285, 290 energy balance, 287, 291 energy density, 19 energy ﬂux, XXII, 19, 127 energy supply, 287, 291 ﬁeld, 283 zeroth-order, 37 ﬁeld quantity, 89 Lagrangean, 157 momenta, 19, 285 momentum balance, 284–286, 290 momentum density Eulerian, XVII Lagrangean, XVII momentum vector, 99 potential, XV scalar potential, XXIII stress, 285 stress-tensor, 290 electromagneto-mechanical interaction, 2, 33 electromechanical interaction, 282 electromotive intensity, XVI, 91 electrorheological approximation, 297, 298 electrorheological eﬀect, XVII, 355 electrorheological ﬂuid, 3, 279, 283, 317, 322 electrorheology, 283 electrostatic polarization mechanism, 280 electrostatic torque, 280 electrostriction, 38 electroviscous eﬀect, 280 elliptical coordinate, XXI end eﬀect, 328 energy (heat) ﬂux, XIX in the statistical formulation, XIX balance, 287 equation, 110 ﬂux, XIX, 77, 130 ﬂux vector, 91 functional, 191 supply, 44, 52, 295 supply density, XX electromagnetic, XX

398

Subject Index

external, XX entropy, 91, 164 ﬂux, 17 ﬂux (Eulerian), XXIII ﬂux vector, 123 inequality, 282, 295, 299 perturbed, 165 production, 17 production density, XXI production inequality, 16 supply, 17 supply density, XX EPS, 357 EPS 3301, 346, 348 equilibrium state, XX, 213 equivalence of models, 89 equivalence of two theories, 89 equivalence relation, 190 ER-eﬀect, 355, 357, 363, 366 ER-ﬂuid Rheobay, 346 ERF Rheobay, 304 Euclidian group, 371 Euclidian transformation, 21, 27, 106, 369 Euler number, XXI Eulerian description, 103 Eulerian formulation, 148 existence, 306 exponent in the power law of the extra stress tensor, XIX extended Lorentz group, XXII, 21 extended thermodynamics, 282 external body couple vector, XVIII extra stress tensor, XVI, XVIII Femlab, 331 ferromagnetic body, 201 material, 3, 164 rod, 217 system, 201, 205 ﬁeld line, 280 ﬁeld strength, 1 magnetic, 1 ﬁnite helix, 247 ﬂat spiral, 250 ﬂexural vibration, 259 ﬂow rate, 329 ﬂuid with transverse isotropy, 281

four-dimensional permutation tensor, 372 four-tensor, 372 Fourier–Bessel series, 264 free energy, 91 functional determinant, 7 functional diﬀerential equation, 139 fusion reactor, 201 Galilei group, 21 Galilei transformation, 21, 27 Gauss’ divergence theorem, 226 generalized viscosity, 283, 332 Gibbs relation, 49, 111, 116, 117, 129 global energy balance, 126 global equation of motion, 270 Hamiltonian, 261, 276 heat conduction, XV, 299 heat conductivity, XXII heat ﬂux, 94 helical coil, 223 helical conductor, 245 helical spiral superconductor, 245 Helmholtz equation, 208 Helmholtz free energy, XXI, 109, 111, 124, 129, 137, 295, 299 high voltage, 282 Hilbert theory, 221 hotness, 89 illumination, 279 impermeability condition, 324 in-plane bending, 251 increment area, 14 length, 14 volume, 14 inextensible ring in bending, 232 inextensible slender ring, 274 inﬁnite helix, 246, 278 inﬁnitely long electrode, 323 instability, 201 insulating oil, 280 intermediate, 150 intermediate conﬁguration, 2 initial, XX intermediate ﬁeld, 151 intermediate state, XVII, 156, 168, 205

Subject Index internal energy, XXI, 94, 285 Legendre transformed, XXII internal energy density, XX Jacobian determinant, XVII jump, XXIII jump condition, 18, 74, 121, 154, 282, 296 energy, 90 for the electromagnetic ﬁelds, 177 momentum, 90 kinetic energy, XX Kirchhoﬀ–Love theory, 259 Kronecker delta, XXI Lagrange density, XVIII Lagrange element, 336 Lagrangean, 261, 276 –Maxwell equation, 104 density, 213, 223 description, 117 electromagnetic ﬁeld, 16 entropy ﬂux, XXIII formulation, 117, 152 free charge, 106 variable, 106, 108, 138 Laplace equation, 3, 208, 260, 321 law Amp`ere, 223, 272 Biot and Savard, 235, 238 Biot and Savart, 229 Faraday, 173 Fourier, 167 Gauss, 152, 173 Gauss–Faraday, 152 Glen, 317 Hooke, 273 Ohm, 167 Legendre transformation, 89, 116, 190, 223 Levi–Civit` a tensor, 12 linear element, 336 linearization, 138, 147 Lorentz model, 152 procedure, 148 linearized perturbed system, 205 long electrodes charged antisymmetrically, 336

399

long range eﬀect, 34 Lorentz force, XVI, 108, 223 formulation, 71, 118, 121, 123, 134, 145, 173, 373 group, 374 invariance, 26 model, 132, 135, 152, 268 stress, 137, 269 transformation, 22 macroscopic description, 281 magnetic anisotropy, 38 dipole, 36 ﬁeld strength, XVII Chu formulations, XVII ﬂux density, XV, 299 induction, XV, 1, 258 auxiliary, XV in the rest frame, XV Lagrangean, XV perturbed, XV interaction energy, 227 interaction integral, 266 permeability, 145, 188 in free space, XXII potential, XXIII, 208, 221, 225 susceptibility, 145, 188 magnetic ﬁeld, 257, 298 uniform transverse, 258 magnetic ﬁeld strength auxiliary, XVII eﬀective, XVII eﬀective, 13 in the rest frame, XVII Lagrangean eﬀective, XVII Maxwell formulations, XVII magnetic ﬁeld strength-electric displacement four-tensor, XXII magnetic permeability, XXII magnetically levitated train, 201 magnetization, 1, 165, 284, 289 magnetization density Chu formulation, XVIII Maxwell formulation, XVIII statistical formulation, XVIII magnetization due to acceleration, 288 magnetization four-tensor, XXII

400

Subject Index

polarization, XXII magnetoelastic, 3 (in)stability, 201 buckling, 201, 202, 204, 207, 211, 214, 217 interaction, 202 model, 205 stability, 202, 206 stability problem, 148 structure, 256 system, XX vibration, 256–258, 271, 276 magnetomotive intensity, 13 magnetostriction, 38, 145 magnetostrictive, 203 Mason number, XIX, 332, 333 material description, 103, 104 electroluminiscent, 279 energy ﬂux vector, 15 ﬂuorescent, 279 Lagrangean formulation, 103 objectivity, 21 phosphorescent, 279 photochromic, 279 smart, 279 thermochromic, 279 with central symmetry, 164 material description Lorentz formulation, 117 Maxwell–Minkowski formulation, 125 statistical formulation, 117 Mathematica, 328 Mathieu function, 209 Maxwell equation, 11, 33, 153, 157, 297 Maxwell formulation, XVIII Maxwell stress, 92, 97, 127 Maxwell–Lorentz aether relation, 24, 105, 107, 118, 372, 374 Maxwell–Minkowski equation, 285, 287, 292, 320 formulation, 55, 117, 125, 134, 173, 197, 224, 226, 271, 284, 297 model, 95, 175, 207, 213, 224, 258, 268, 272, 375 measurable quantity, 35 mesostructure, 281 metric tensor, 372

microstructure, 281 mineral oil, 280 Minkowski ﬁeld, 92 Minkowski formulation, 93, 372 Lagrangean version, 133 Minkowskian electric ﬁeld strength, 284 ﬁeld magnetic strength, 284 magnetization, 373 polarization, 373 mixture, 4 MKSA-unit, 36 model I, 42 model II, 51 model III, 55 model IV, 66, 176 model V, 71 modiﬁed internal energy, 294 moment of inertia, XVII, 216 moment of momentum, 97 momentum equation, 290 MRI-scanner, 201, 256 n parallel rods, 278 Newtonian, 327 behaviour, 307, 317 non-conducting current, 13 ﬂuid, 311 non-relativistic, 24 approximation, 92 electromagnetic energy balance, 291 electromagnetic energy supply, 291 non-uniform electric ﬁeld, 281 non-viscometric, 366 norm, XXIII normal stress diﬀerence, XIX eﬀect, 282, 310 objective axial tensor, 370 axial vector, 371 polar vector, 370 oblique electrode, 365 observable quantity, 35 Ohmian conductor, 304 Onsager relation, 79, 132 order symbol, XXIII

Subject Index orthogonal matrix, XX out-of-plane bending, 234 paraﬃn, 280 parallel rods, 237 superconducting rods, 242 tori, 234 particle chain, 280 particle polarization, 280 Peltier heat, 82 permanent magnetic electrode, 288 permutation tensor four-dimensional, XVI three-dimensional, XVI perturbation on the equilibrium state, 213 perturbed entropy density, XX perturbed state, XVI, 172 Pfaﬃan form, 49 photon radiation, 279 photonic crystal, 282 piezoelectric coupling, 202 piezoelectric vibration, 257 Piola–Kirchhoﬀ stress, 165 plate, XIX, 2, 264 clamped, 263 current-carrying, 203 ferromagnetic, 202, 204, 264 magnetic, 203 piezoelectric, 202 rigidity, XVI simply supported, 264 superconducting, 203 thin soft ferromagnetic circular, 258 plate conduction model, 281 Poiseuille type, 359 Poisson equation, 321 number, XXII polarizable, 279 polarizable material, 282 polarization, 1, 165, 284 polarization density, XIX Chu formulation, XIX Lagrangean, XIX Maxwell formulation, XIX perturbed, XIX statistical formulation, XIX

polarization gradient, 88 polarized dielectric, 298 polyelectrolyte, 279 polynomial expression, 188 polynomial representation, 140 power law, 305 constitutive relation, XVIII model, 283, 317, 328 alternative, 333, 346 pressure, XIX drop, 328, 342, 358 gradient, 359 principle of virtual power, 87 protruding electrode, 365, 366 pseudoplastic behaviour, 346 quadratic element, 336 quasi-static process, 271 rate of strain tensor, XVI, 9, 299 Rayleigh quotient, 262, 276 Rayleigh’s principle, 202 reactor, 223 reduced entropy inequality, 124 regularization parameter, XXII relaxation time, 279 residual inequality, 130 rest frame, 25 retreating electrode, 365, 366 Reynolds number, XX, 332, 333 Rheobay, 352, 357, 359 Rheobay TP AI 3565, 310, 334 rigid-body ﬁeld, 271 state, XV, XVI, 2, 151, 159, 168 translation, 260 ring equation, 234 rotational viscometer, 318 scanner, 223 sedimentation, 279 semi-relativistic, 24 shape memory alloy, 279 shear force, 266, 270 shear modulus, XVII elastic, XVII shear stress Bingham, XXIII Casson, XXIII

401

402

Subject Index

Casson-like model, XXIII shell current-carrying, 203 magnetic, 203 shock absorber, 282 shock wave, 18 short range eﬀect, 34 silicone oil, 280 simply supported rod, 231 singular surface, XXII singularity, 308 slender rod-like structure, 236 slenderness parameter, XXI smart ink, 282 smooth electrode, 367 soft ferromagnetic beam, 207, 214 soft ferromagnetic parallel rod, 217 solid magnetizable, 26 polarizable, 26 spatial description, 36 spatial velocity gradient, 9 speciﬁc body couple, XVIII heat, XV, 145 internal energy, 113 speed of displacement, XXI of light, XV of propagation, XXI spin tensor, 9 spiral, 223 splines, 248 stability, 150, 280 of weak and strong solutions, 306 statistical formulation, 66, 123 Chu, XVI Lorentz, XVI Maxwell, XVI statistical model, 132, 373 strain tensor Eulerian, XVI Finger, XVI Green, XVI Lagrangean, XVI stream line, 366 stress tensor, XX extra Cauchy, XX

extra Cauchy–Maxwell, XX ﬁrst Piola–Kirchhoﬀ, XX, 15, 110 perturbed, XX Maxwell, XVII, 19, 136 perturbed, XX Piola–Kirchhoﬀ, 163, 166 second Piola–Kirchhoﬀ, XX, 110, 114 stretching, XVI, 9 stretching tensor, 352 strongly inhomogeneous electric ﬁeld, 349 Strouhal number, 289 superconducting, 3 coil, 201, 204 generator, 201 ring, 271 rod, 227 solenoid magnet, 203 structure, 214, 222, 224, 236, 256, 276 susceptibility electric fourth-order, XXIII second-order, XXIII magnetic fourth-order, XXIII second-order, XXIII suspension, 279 symmetric bending, 262 symmetrical buckling mode, 221 Taylor series expansion, 140, 143, 146 theory of complex functions, 215 thermodynamic equilibrium, 301 potential, XVII pressure, 300 thermoelastic coupling tensor, XXII thermostatic equilibrium, 77, 130 thickness to width ratio, XXI Thomson heat, 82 three-dimensional potential, 215 toroidal coil, 223 total current, 232 total electromagnetic couple, XVIII total electromechanical balance law mass, 297 momentum, 297 traction, XX

Subject Index transformation electromagnetic variables, 101 group, 370 transport theorem, 11 transverse displacement, XXI transverse magnetic ﬁeld, 214 two concentric tori, 235, 236 two parallel rings, 278 two parallel rods, 278 two–dipole model, 39, 104, 374 nonsymmetric stress tensor, 42 symmetric stress tensor, 51 uniqueness, 306 unit normal, 10 unit normal vector Eulerian, XIX Lagrangean, XIX vacuum, 161 valve, 282 variational formulation, 3 variational method, XXI, 201, 211, 216, 227, 235, 244

403

variational principle, 204, 211, 220, 224, 235, 276 for magnetoelastic vibrations of superconducting structures, 276 vectorial surface element, XV, XVI vegetable oil, 280 velocity modulus, 359 vibration, 201 viscometric, 330 ﬂow, 308 viscosity, 279 bi-viscosity model, XXII generalized, XXII viscous magnetizable, 99 polarizable, 99 viscous ﬂuid, 299 volume element, XVI von-K´ arm´ an theory, 257 vorticity, 9 Wiener–Hopf technique, 330 yield stress, XXIII Young’s modulus, XVI

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